On the Theory of Schur-Rings.Begriff der Kategorie zusammen. Daher qvird [i2r die Schur-Ringe...

On the Theory of Schur-Rings.

OLAF TAMASCHKE (Tfibingen, Germania Occid.)

Zusammenfassung. - Diese Arbeit versueht, die yon lssai Schur [1] e~tdcc~le ~q~d yon Wie. landt ([14], [15], [16], [17]) betrdchtlieh n'eitere~tn~ickelte Methode zur Untersuchu~g vo~ endlichen Perm~tationsgruppen zn eiq~er Theorie der Schur.Riq~ge zu entfalten. Der Grundgedanke ist sehr einfach: Die Schur.Riq~ge qverden nicht als eine sjoezielle Klosse yon Ri~gen aufgefa~t, sondern als ei~e eigeq~e mathematische Struktur. Araeh unserer heutigen Ansieht fallt der Begriff der mathematischen Struktur qveitgehe~d mit dem Begriff der Kategorie zusammen. Daher qvird [i2r die Schur-Ringe (genauer: fiir die Schur.Algebren) ein eigener Homomo~Thiebegriff (Definition 1.5) eingefi~hrt, der eine Ka. tegorie liefert (Theorem 1.6).

Ein weiterer Leitgedanke ist mit dem kategoriellen Grundgedanken sehr eng vet. knii~pft. Die Theorie der Sehur-Ri~ge qvird als eine Verallgemeineru~g der Theorie der endlichen Gruppen a~¢fgefa~t und in diesem Siq~e eq~tqvickelt. Dabei ist die Theorie der endIichen Gruppen ve~mittelst der Gruppe~ringe der e~,dlichen Gruppen (die eiq~e spe. zielle Teilkalegorie der Kategorie aller Schur-Ringe siq~d) in die Theorie der Schur-Riq~ge eingefi~gt. Hierfiir ist es wichtig, da.~ die Morphismen der Gruppen~i~ge in der Kategorie der Sch~fr.Ringe genau die yon den Gruppenhomo~uo~phi~men induzierten Gr~ppen= ringhomomorphismen sind.

Die ~'inbettuq~g tier Theorie der e~dlichen Gruppen in die Theorie der Sehur.Ri~ge vollzieht slch entlang dreier Entwicklungslinien.

D~e erste ist eine verallgemeinerte Charaktereq~theorie ([2], [3], [5], [6], [7] und [8]), die die.'.Theorie der (geqv6h~liche~) Ct~araktere yon e~dlichen Gruppe~ ats Spezialfatl ent~ilt.

Die zweite ist die Verk~i~fung der Sti'~tkt*~r jedes Schur.Ringes T a~f einer end. liche~, Gr,ppe G ~qit gewissen Klassen yon Untergruppeq~ yon G. Es rverden die Be- gri/fe der T. Unlergr~ppe (Absehq~itt 3}, des T-Normalteilers (Abschnitt 4), ~nd der T.sttbnormalen Untergr~ppe (Abschq~itt 8) eingefi~hrt. Die T.Untergruppen bilden einen Teilverband des Verbandes aller Untergruppen yon G (Theorem 3.4). Die T.Normalteiler sind ~'e~au die Kerne (Definition 6.1) tier Homomorphismen der Schur.Algebren QT (Theoreme 6 2 und 6.3).

Der dritte und rvohl zugleich der ~vichtigste Aspekt ist die Gi~ltigkeit des Homomof phiesotzes (Theorem 6.2) u~d der Isomo~phiesdtze (Theore~ne 7.1 und 7.2) fi2r Sehur-Al. gebren. Au f diese Satze gri2ndet sich der Vier. Untergr~ppen.Satz (Zassenhaus' Len~ma; Theorem 9.1), der den Veq'feineruq~gssatz fi~r T-Subnormalketten (Theorem 9.2) und den Jordan-HSlder ~atz [i2r T-Kompositivnskellen (Theorem 10.3) q~ach sich zieht.

Als die Thevrie der Schur-Ringe uq, gefahr den soeben geschilderten Stand erreicht hatte, tauehte die Idee a~f, diese '_rheorie au f beliebige Gruppen zu verallgemeinern ([9], [10], [11], [12], [13]). Das fi~hrte zum Begriff der Schnr-Halbgrupl)e (Definition 1.9). Der zugeh6rige HoqnomoqThiebegriff (Definition 1.11) liefert die Kategorie aller Schur- Italbgruppen (Theorem 1.12), die die Kategorie aller Gruppen als echte Teilkategorie enthdlt. Jedem Schur-Ring T i2ber einer endlichen Gruppe G ~cird eine Schur-Halb- gruppe T i~ber G zugeordnet (Theorem 1.15). Jedem Homomorphismus ¢~ einer Schur- Algebra f i t fiber G wird ein Homomorphismus q> yon T z,~geordnet (Theorem 1.16). Das Paar der Zuordnungen ~ T ~ T, ~ ~ ¢ ist ein Funktor au[ der Kategorie aller Schur-Algebren in die Kategorie aller Schur-Halbgr.uppen i~ber endlichen Gruppen (Theorem 1.17).

Annati di Matematica !

O. TAMASCHKE: On the Theory o] Schur-Rings

The method of Schur, introduced into the theory of finite permutat ion groups in 1933 by ISSAI ScrruR's paper [1], consists in the investigation of a certain class of subrings of tile group rings of finite groups which Wllg- LA~DT called S-rings, i.e. Schur-riugs ([17], Definition 23.1). Compared with character theory, which still dominates many parts of the theory of finite permutat ion groups, it is much more elementary, and yet, in certain in- stances, it is no less successful than its powerful rival. Apart from Sc•UR's original idea this is chiefly due to tl:e work of WIELA~DT ([14], [15], [16], and [17], Chapter IV).

Already in the early days of the theory of Schur and Wielandt underlying connections with char,~cter theory had been worked out arid used in applications by WIELANDT ([16] and [17], Chapter V), But a new branch of this theory developed ([2] - [8]) when, by turning away from the e lementary fea. tures of S-rings, attention was focused on the semi-simplici ty of the S-algebras over the field C of complex numbers (a fact that was well known for a long time. cf. [15], p. 386, foot-note). Investigations in that direction led to the discovery that the coefficients of the irreducible representat ions of an S-algebra (over C) on a finite group G can be e~pressed in terms of the ir- red~cible representat ions of the group G itself [6]. From there it was only one fur ther step to discover that every S-r ing on a finite group G defines its own character theory on G [7]. That theory incorporates ordinary character theory as a special case by appl~qng it to the group ring ZG qua S-ring. Though this generalized character theory still lacks any spectacular application (an application cf minor importance is given in [4]) it has had one re- markable sequel so far. A generalization of normal subgroups arose from it [8]. Each S- r ing T on a finite group G defines a class of T-normal subgroups of G which are characterized by those S-r ings on G that are two- sided ideals of T. So the detour via the generalized character theory at least led to a fundamenta l notion of S- r ing theory whose definilion is free of character theoretical ingredients.

The concept of T-normal subgroup became the starting point for yet another development which embraces both the e lementary and the character theoretical part of the theory of S-rings. It is the purpose of this paper to present this new approach to S- r ing theory. The b:~sic idea is quite simple. The S- r ing is considered not as a special type of ring but as a mathematical s tructure in its own right. This means that a notion of S- r ing homomorphism has to be produced so as to obtain a category. It turns out (cf. the remarks after Definition 1.3) that a slight modification is necessary. It is not the s t ructure of an S- r ing but the s tructure of an S-algebra over a field of characteris t ic zero (Definition 1.4; |he field Q of rational numbers is sufficient) that has to be used in order to obtain a workable notion of homo. morphism (Definition 1.5) and hence a category (Theorem 1.6). The kernels

O. TAMASCHKE: On the Theory of Schur-Rings 3

of the homomorphisms of an~,S-algebra [QT on a finite group G (Definition 6.1) are exactly the T-normal subgroups of G (Theorems 6.2 and 6.3).

We consider S-r ing theory as a generalization of finite group theory where finite group theory is incorporated via the group rings qua S-rings. To rea- lize this programme we introduce notions which link the S-r ing structure with the structure of the underlying group. Apart from the generalized character theory already mentioned above this is primarily done by defining several classes of subgroups, namely the T-subgroups (Section 3), the T-normal subgroups (Section 4), and the T-subnormal subgroups (Section 8). Inter alia, the T-subgroups form a sublattice of the lattice of all subgroups of G (Theo- rem 3.4). Every homomorphism ¢~ of an S-algebra QT on G into an S-algebra QS on a finite group F relates T-subgroups of G to S-subgroups of F (Sec- tion 5) and yields a lattice monomorphism of the lattice of all those T-subgroups of G containing the kernel of .~ (Theorem 6.7).

Of the lattice theoretical properties of the T-normal subgroups we only know that the product of any two of them is again T-normal (Theorem 4.8), and of the T-subnormal subgroups we are able to prove that the intersection of any two of them is aga in T-subnormal (Theorem 8.5).

Apart from the generalized character theory and the various classes of subgroups it is the S-r ing version of the Homomorphism Theorem (6.2) and of both Isomorphism Theorems (7.1 and 7.2)which supports the con- ception of S-r ing theory as a generalization of finite group theory.

In consequence of these theorems tho S-r ing version of the Four Sub. group Theorem 9.1 (Zassenhaus' Lemma) can be proved which in turn implies the following results. As the factors of a T-subnormal chain we define certain S-r ings on the subgroups of that chain which are connected with T. The Refinement Theorem for T-subnormal chains (9.2) and the Jordan-HOlder Theorem for T-composition chains (10.3) hold with S-algebra isomorphisms of the factors

Once that state of the theory of S-r ings was achieved the idea of generali- zing the whole theory from finite to arbitrary groups broke through. The result was the concept of an S-semigroup on a group G ([.9] and [10]). With its own notion of S-semigroup homomorphism (modelled after the S-algebra homomorphism) a category of S-semigroups is obtained which properly contains the category of all groups. The kernels of the homomorphisms of an S-semigroup T on G are the T-normal subgroups of G. With |hese concepts of homomorphism, kernel, and T-normal subgroup the S-semigroup versions of the Homomorphism Theorem, the Isomorphism Theorems, the Four Sub. group Theorem, the Jordan-H6tder Theorem for T-composit ion chains and other statements hold as proper generalizations of the relevant statements of group theory ([9], [10], [I.1], [12]; an even more recent development of these ideas can be found in [13]).

O. TAMASCHKE: On the Theory of Schur-Rings

Every S-r ing T on a finite group G is associated with an S-semigroup T on G (Theorem 1.15), every S-algebr~ homomorphism ~ is associated with an S-semigroup homomorphism (!) (Theorem 1.16), and the pair of mappings T ---~,-T, ~--~ (I) is a functor on the category of all S-r ings into the category of all S-semigroups (Theorem 1.17).

1. - S-Rings, S-Algebras, S-Semigroups.

Let G be a group. For any set X ~ G we write

x =

If X is a finite set, then we form the element

X ' = Y ~ g - - g e X

in the group ring ZG.

DEFINITION 1.I. (WIELANDT [[7], 23.1). - Let G be a finite group. A sub. ring T of the group ring ZG is called an S-r ing on G i f there exists a set • . of non-empty subsets of G such that the following hold.

(i) G -- U ~;. ' g e ~ :

(2) $ - - - ~ or $N~- '9~ for all S,¢gE~.

(3) ~_1¢ ~. for all ¢G e ~..

(4) .:---- E z~,~g~___ with z~,-geZ for all ~e T.

(5) The ring T is generated by the elements ¢g. ~ ¢ ~ (which conse-

quently are a Z-basis of T).

The set ~ is uniquely determined by the axioms (1)-(5). Therefore we call the elements of • the T-classes of G. The ~elements ~ , ~ ~ ~, of the group ring ZG are called the T-class sums of G.

The T-class ~ of G which contains the unit element 1 ~G is a subgroup of G, and

~ ¢ g ~ - - ~ l z : t¢~1 '~ holds for all z c T

[15], (1.6}. It follows that

0. TAMASCHKE: Or/ the Theory of Schur-Rings 5

EXAmPLeS 1 . 2 . -

I) The gro~p ring 2G of any finite group G is an S- r ing on G with ~ - - G .

2) The centre ~(ZG) of the group r i n g [ Z G is an S- r ing on G with -- the set of all eonjugacy classes.

3) Let H be a subgroup of the finite group G. Then the subring G / / H of the group ring ZG which is generated by the double coset sums HgH, g e G , is an S- r ing on G wi~h ~ . -=IHgHlgeG} . We call G / / H the

double coset S-r ing of G modulo H. With that notation we have ) .G= G / / t .

Let 17 be a finite group, S an S- r ing on F, and ~ the set of a l l~S-c l a s ses of F (i. e. ® plays the same r61e for S as IZ does for T).

DEHNI~mN 1.3. - A mapping ~ of T into S is called a homomorphism of the S-r ing T on G into the S-r ing S on F i f the following hold.

(1) (ax)~ - - ~z9 foJ' all v,'c e T.

(2) (~ -{- ":)~ = ~ -b ~ for all ~,~ e T.

(3) For every T-class ¢g of G there exists an S-class S of F such that

IS I ' ~ - I ~ I ' S and iS _i[.C¢g_~)~__t,~-1i.8 - t .

Proper t ies 1.3 ( t ) a n d (2)of an S- r ing homomorphism ~ simply mean that ~ is a ring homomorphism, and this does not need any further discus- sion. Proper ty 1.3 (3) means the compatibil i ty of ¢g with the "S - s t ruc tu re" of the S-r ings T and S, and the formulation we have chosen may need some motivation. We want our theory as close to group theory as possible. Every homomorphism :¢ of the group G into the group F induces a homomorphism ¢¢~ of the group ring ZG into the group ring Z F in the natural way. For any subgroup H o[ G, and any g e G the homomorphism ~ maps double cose onto double coset, namely

(HgH) ~ ~- H~g~H ~,,

but for ¢p~ we have the equation

!Hag ~H~ I" (HgH)~ ~- I HgHI . H~g~H ~ .

So if we want any group homomorphism a to induce an S- r ing homomorphism of the double coset S-r ing G / / H onto the double coset S- r ing G~//H ~ we have to choose the formulation 1.3 (3).

But there is an aggravating deficiency with Definition 1.3.

O. TAMASCHKE: On the Theory o] Schur-Rings

It is true that the product of two S-r ing homomorphisms again is an S- r ing homomorphism in the sense of Definition 1.3. ]~u~ what makes Defi- nition 1.3 deficient is the fact that it does not yield enough S-r ing homomorphisms. For any given S- r ings T on G and S on F there should exist at least one homomorphism of T into S. This is not guaranteed by Defini- tion 1.3, for 1.3 (3) can hold only if the positive integer ISI divides the positive integer I~l.

However this deficiency can easily be remedied. If we consider the equations of 1.3 (3) in the group algebra QG (Q lhe field of rational numbers) then they may have enough "solutions".

Now we are going to modify our concepts so as to obtain enough homomorphisms.

Let ~2 always denote a subfield of the field C of complex numbers. In this section and in the Sections 3-10 it would suffice to take ~ - - Q . In the next section we shall be concerned with ~ - ~ C. Thereh~re we do not specify ~2. We could even subst i tute ~ by any other field of characterist ic z e r o .

To generalize even further and take any field will cause some problems. But we do not want to investigate these questions in this paper.

DEFI~I~O~ 1.4. - Let T be an S - r ing on the finite group G. Then the sub- algebra

l'~T: = { E a~g~la`ge~2 } ' g e e - -

of the group algebra ~2G which consists of the linear combinations of the T-class sums with coefficients from ~2 is called an S-a lgebra on G over ~2.

Now we rephrase Definition 1.3 for S-algebras.

DEFI~i~ION 1.5. - Let T be an S - r ing on the finite group G, and S an S - r ing on the finite group F. A mapping ~ o f ~2T into ~2S is called a homomorphism of the S-algebra ~2T into the S-algebra ~2S i f the following hold.

(I) ( ~ 2 -- ~v:v for all ~,~ E ~2 T.

(2) (a Jr ~)~ -- ~, + ~ and (a:) ~, ---- az~, for all ~,~ ~ ~2 T and all a e ~2.

(3) For every T-class ~ of G there exists an S-class $ on F such that

1 S and 15_11

A homomorphism ? of an S-algebra ~ T into an S-a lgebra ~ S is called an epimorphism, a monomorphism, or an isomorphism if ? is a surjeetive, an injeotive, or a bijeotive mapping respectively. These definitions of epimor-

O. TAMASCHKE: On the Theory o/Schur-Rings

phism and monomorphism are naive ones. Whether they have the same meaning in the categorical sense will have to be discussed later

Definit ions 1.4 and 1.5 now yield the desired result.

T]~EOREM 16. - The class of all S-algebras on finite groups over ~2 toge. ther with their homomorphisms (in the sense of Definition 1.5) form a category.

PRonto. -

I. Let T be an S- r ing on the finite group G, and S an S- r ing on the finite group F. We denote by g~ the S-c lass of /r containing the unit element of F. The linear extension onto f~T of the mapping

1 1 i~ I. ~.---~ ~ -~ .g j for all ¢geff~

is a homomorphism of 12T into f lS in the sense of Definition 1.5. We call it the zero homomorphism of f~T into ~2S. Therefore the set

I Ioma(T,S)

of all homomorphisms of the S-a lgebra ~2T into the S-a lgebra f~S is non- empty for any two S-r ings T on G and S on /7. If T' is an S- r ing on G', and S' is an S- r ing on F', then

(T,S) @:(T',S'j implies HomdT, S)flItoma(T',S') = ¢.

II. For each triple of S-r ings

T o n O, S o n F, R o n E,

and each pair of homomorphisms

~ Homa(T,S), ,b ~ Homa(S,R)

their product as mappings is a homomorphism, i.e.

~,~ e Homa(T,R),

and the multiplication of homomorphisms is associative.

II[ . The identity mapping tot of any S-a lgebra ~2T on G is a homo- morpbism of OT, and for every ~: e I~oma(T,S) we have

inT~ = ¢p = ~ias.

8 0. TAMASCHKE: On the Theory o] Schur-Rings

I - I I I show that the propert ies of a category are satisfied, and our Theo- rem 1.6 is proved.

We mention two important subeategories of the category of all S-a lge . bras over ~ . For any double coset S - r i n g G//H we call the S -a lgebra ~2G//H: -- ~2(G//H) the double coset S-algebra of G modulo H over ~2.

TI4EOREM 1.7. -- The class of all double eoset S-algebras over ~2 together ~vith their homomorphisms (in the sense of Definition 1.5) form a proper sub- category of the category of all S-algebras over ~2.

The proof of Theorem 1.7 is l i terally the same as that of Theorem 1.6. But we have to issue a warning. From the results of our later sections it is easy to deduce that the homomorphic image of a double coset S-algebra is a double eoset S -a lgeb ra if a homomorphism is given into a double coset S-algebra. But there exist double coset S -a lgebras isomorphic (in the sense of Definition 1 .5) to S -a lgebras which are not double coset S -a lgebras themselves. This means that the property of being a double c0set S -a lgeb ra is not preserved by S -a lgeb ra isomorphisms. This fact follows from [17 l, Theorem 24.1. There the transit ivity module of the stabilzer G1 of a letter in a transitive permutat ion group G with a regular subgroup is an S - r ing which - - a p a r t f r o m trivial e x c e p t i o n s - - is not a double coset S- r ing , and yet its S -a lgebra over ~ is isomorphic to the double coset S -a lgebra ~GIIG, .

THEOBEM 1 . 8 . - The class of all group algebras of finite groups over ~2 (qua S-algebras over ~2) together with their homomorphis~ns (in the sense of Definition 1.5) form a proper subeategory of the category of all double coset S-algebras over ~2.

The proof is clear. Let ~0 be a homomorphism of the group algebra ~2G qua S-a lgebra on G into the group algebra ~/~ qua S -a lgeb ra on F. Then ~ ¢ : g ~ g v is a homomorphism of the group G into the group F. Con- versely, if ~. is a homomorphism of the group G into the group F, then the l inear extension :c a of a onto ~2G is a homomorphism in the sense of Deft. nition 1.5 of ~2G as S -a lgebra on G into ~2F as S -a lgebra on F. Therefore G ~ ~2G is a o n e - t o - o n e correspondence between the finite groups and their group algebras qua S-a lgebras , and : ¢ ~ ~a is a o n e - t o - o n e correspondence between the group homomorphisms and the S -a lgebra homomorphisms of the group algebras of all finite groups over ~l. The pair of mappings G - ~ I 2 G and a ~ ~a is a functor of the category of all finite groups onto the category of all group algebras of finite groups over ~ as S-a lgebras . That way the theory of finite groups is embedded into the theory of S-a lgebras .

W e started with S-r ings , and went over to the r icher s tructure of S-algebras over ~2. Now we go in the opposite direction, and go over to a poorer

O. TAMASCHKE: On the Theory o/Schur-Rings 9

s t ructure namely to S-semigroups which have been modelled on the concept of S-rings. The poorer s t ructure has wider applications. It is not bound to finite groups. So let G be an arbi t rary group (not necessari ly finite). The set

is a semigroup with respect to the subset (or "complex") multiplication

( X , ] : ) - - ~ X Y : = { x y ! x e X and y e Y}.

DEFINI~ION 1.9. ([t0], p. 74). -- A subsemigroup T of G is called an

S-semigroup on G i f it has a uni t element and i f there exists a set ~ . ~ G such that

(1) G = U ~ .

(2) S = ~ or S ~ " g = ~ for all $,¢gEff..

(3) 'i~ - 1 E ~ for all ~ E ~..

(4) X = U ~ for all X ET.

~nx4~

(5) T is generated by ~., that is every element of T is the product of a finite number of elements of ~..

The set ~ is uniquely determined by the axioms (1)-(5). Therefore we call the elements of ~: the T-classes of G.

EXAMPLES 1 .10 . -

1) Every group G is an S-semigroup on G with fZ----- G.

2) The semigroup (with respect to complex multiplication) that is generated by taking ~: to be the set of all conjugacy classes, is an S-semigroup on G.

3) Let H be a subgroup of the group G. Then the semigroup G / H (with respect to complex multiplication) which is generated by ~: ~ {HgHig E G} is an S-semigroup on G. We call G/H the double coset S-semigroup of G modulo H.

Let F be a group, ~ an S-semigroup on F, and ~ the set of all ~-classes of F.

DEFINITION 1.11 ([10], Definition 2.t). - A mapping • of T into ~. is called a homomorphism of the S-semigroup T on G into the S-semigroup ~. on F i f the following hold.

Annali di Matematica 2

l0 O. TAMASCHKE: Or/ the Theory of Schur-Rings

(1) (XY)¢--- -Xcy ¢ for all X, Y E T .

(2) X ¢ ~ U ¢~¢ for all X ET.

(3) For every T-class '~ of G there exists a Z-class $ of F such that

~¢ ~ $ and (~-~)¢ -~ $-~.

To speak of the mathematical s t ructure of an S-semigroup means the following.

TttEOREM 1.12. - The class of all S-semigroups together with their homomorphisms (in the sense of Definilion 1.I1) form a category.

PRooF. - For any two S-semigroups T on G and ~ on F the mapping

of T into E which maps every element X E T onto the Z-class S~ containing the unit element of /P is a homomorphism of T into Z in the sense of De- finition 1.11 (note that S~ is a subgroup of F and also the unit element of

by [10], Lemma 1.2). We call it the zero homomorphism of T into E. Other- wise the proof of Theorem 1.12 is completely analogous to the proof of Theorem 1.6.

THEORE~ 1.13. -- The clcms of all double coset S-semigroups together with their homomorphisms (in the sense of Definition 1.11) form a proper subcate. gory of the category of all S-semigroups.

The remarks after Theorem 1.7 also apply analogously to the double eoset S-semigroups and S-semigroups as to the double coset S -a lgebras and S-a lgebras .

TEEORE~ t.14. - The category of all groups is a proper subcalegory of the category of all double coset S-semigroups.

The group homomorphisms of a group G and the S - semigroup homomorphisms of G qua S-semigroup are one and the same. Therefore the whole of group theory is part of S - semigroup theory.

For the remainder of this section we study the relationship between S-r ings and S-a lgebras on one side and S-semigroups on finite groups on the other.

T~]~onEM 1.15. - Let G be a finite group, T an S-r ing on G, and ~ the set of

all T-classes of G. Then the subsemigroup T of G which is generated by ft. (i.e. every element of T is the product of a fissile number of elements of ft,) is an S-semi.

O. TAMASCHKE: On the Theory o/Schur-Rings 11

group on G with ft. as the set of all T-classes of G.

We call T the S-semigroup related to the S-ring T.

PROOF. 1.9(1)-(3) and 1.1(1)-(3) are identical, and 1.9(5) is satisfied by the definition of T. Under these conditions 1.9(4) is equivalent to the state- raent that for any ~,~; 'E~ their complex product ~ ' is the set theoretical union of elements of ~ . By 1.1(4) we have

rg ¢6' ~-- Y, a% ~;,, ~7,,¢g ''_ with 0 ~ a,g, ~;, ~;,, E Z.

It is clear that g E ~ ' if and only if a%~, ,~ ; , ,>0 for the unique '~"EIZ

such that g E ~ " holds. Hence

g ~ ' = U ~", :g"etZ

aW, W', W" > o

and our theorem is proved. Moreover, our proof shows that the mapping

a, ~> o

is a semigroup homomorphism of that subsemigroup of the multiplicative semigroup of the ring T which is generated by the T-class sums ~g, rg E ~, onto T.

T~EOREM 1.16. - Let ~ be a homomorphism of the S-algebra ~ T on the finite group G into the S-algebra ~2S on the finite group F. Then there exists a unique homomorphism ~ of the S-semigroup T on G related to T into the S-semigroup E on F related to S such that

( ] - ~ . ~ ) ~ - ( i - ~ ¢ ~ . ~ ¢) f o r a l t ~ E ~ .

We call • the S-semigroup homomorphism related to the S-algebra homomorphism ~.

PaOOF. - For every T-class ~ of G there exists (by Definition 1.5(3;) an S-class ~ of F such that

1

12 O, TAMASCHKE: Or/ the Theory o/Schur-Rings

and we set

We define

' ~ * . = $ .

X¢ " ~--~- LJ

~ x

~ ¢ for all X 6 T .

Thus; O:X--*X ¢ is a mapping which satisfies (2) and (3) of Definition 1.11. Take any X, Y 6 T , and set

1 .¢~'

With the equations

(-~ ~ ) ( ~-~ " ¢g' ) = Y, d ~, ~;,, ~7, , 1 [" ~,'o~ ~ i " ~' ' '

which hold by Definition 1.4, we obtain

~ = z ( z d~,v,,~,,)]~i.~;"= z zv"~l .¢g,, tp~ ~;=x_ ~; =xY

~'c_ y

with positive rationals z~,,. Since ~,~:E~T, the S-a lgebra homomorphism

yields

~+~i~;¢i - 1 ~ v + ~ " ~;'¢ = ~ = ( ~ ) ~ = ~ x ~'~ Y

1 = ~, z?;,, - - .~"+ ~,,~ I~"+l

It follows that X c Y ¢ =(X:Y) ¢, and hence 1.11(i) is satisfied as well. Therefo- re (I) is a homomorphism of the S-semigroup T into the S-semigroup 2. It certainly has the stated uniqueness property since every homomorphism of the S-semigroup T is uniquely determined by the images of the T-classes.

Tt~EOREM 1.17. - The pa ir of mappings

~2 T ~ T (defined by Theorem 1.15)

~ ¢ (defined by Theorem 1.16)

O. TAMASCHKE: On the Theory o] Schur-Rings 13

is a covariant functor on the category of all S-algebras on finite groups over into the category of all S-semigroups on finite groups.

PROOF. - For ~----ior~--~ the identity mapping of ~2T, we obtain ( I ) - iv the identity mapping of T. Let a triple of S-algebras

~2T on G, f~S on F, f~R on E

be given with the related S-semigroups

and let

T on G, ~ on F, P on E,

be a pair of S-algebra homomorphisms with the related S-semigroup~homomorphisms

( I ) : T ~ , ~ : ~ P .

Then

1 . ( ~ ¢ ) , F : ( 1 )+ (1 )~+

holds for every ~E ~ Therefore the S-semigroup homomorphism

¢ ~ : T ~ P

is related to the S-algebra homomorphism

++:t2T-- ~2R,

and our theorem is proved., If we apply Theorem 1.15 to the double coset S-algebras, then the S-se-

migroup related to a double coset S-algebra ~2G//H obviously i8 the double eoset S-semigroup G/H.

TI~EOI~EM 1.18. - The pair of mappings

~2G//H--* G/H (defined by Theorem 1.15)

~ ~9 (defined by Theorem 1.16)

is a eovariant funotor on the category of all double coset S-algebras on finite groups over ~2 onto the category of all double coset S-semigroups on finite groups.

14 O. TAMASCHKE: On the Theory o] Schur-Rings

If we apply the functor of Theorem 1.17 to the group algebras of finite groups over Ft qua S-algebras, then we obtain the inverse functor t 2 G ~ G, ~n~:¢ of the functor mentioned after Theorem 1.8.

An important question remains unanswered by this paper. Is the func. for of Theorem 1.17 a functor onto the category of all S-semigroups on finite groups? We re-word that quest ion as

PROBLEI~ 1.19. - Let T be an S-semigroup on a finite group G with Ig as the set of all T-classes of G. Is the ring T generated by all T -c lass sums ~ , cg E ~, an S - r i n g on G, i.e. is the product of any two T-c lass sums always

a linear combination of T-c lass sums with (necessarily) non-~egalive integral coefficients? I n other words, given a decomposition ~. of a finite group G such that ft. is invar iant by taking inverses (i.e. 1.1(1)-(3) hold), does

always imply

~ ' = U ~" for all ~ , ~ ' E W'~ig

g , , fl g g' :Ji: ~3

¢g~' = ~. a% g,, ~;,, eg" with O <= a~;, g,, g,, E Z for all ¢g, cg' 6 ~ ? g;,,~_g. ---

2 . - T-Characters .

From now on we keep the following notations throughout this paper.

G is a finite group.

T is an S - r i n g on G.

T is the S-semigroup on G related to T (Theorem 1.15).

= {~i, . . , ~t} is the set of T-classes ( = T-classes) of G.

ti-----I~il the number of elements in the i - t h T-class of G.

z t - - - -~ ( i = 1 , ..., t) the T-class sums of G. _ o _

Under the assumption of 1.1(1)-(3), (5) condition (4) of the definition of an S - r ing is equivalent to the statement that all products

t

• i~j = v a, ik~ (i, j = 1, ..., t) k = x

of T-c lass sums are l inear combinations of the T-class sums with (necessarily) non-nega t ive integers a~ik as coefficients. For every T-c lass '~i there exists a T-c lass ~;j such that ~ = '~i (1.1(3)). We write i* for that j , i.e.

'g~, : = %'# ( i = 1, . . . , t).

O. TAMASCHKE: OF/ the Theory o/Schur-Rings i5

For every

= x , ~ ~ x g G ,

the average of X is defined in the group algebra QG as

1 -IxI .x_.

If H is a subgroup of G, then the subgroup average

is all idempotent of the group algebra QG. In Q T we have the~equations

t

aqktk

aqe " ~ tdi

(i, j , k = 1, ..., t)

1 for the T-class averages ~ = ~._:~ (i = 1 , ..., t).

We choose the notation such that ~1 is the T-class containing the unit

element of G. Then x~ is the unit element of QT~ and also of CT. For any element

T = Z %g ~% E C) geG

of the group algebra CG we form the element

Y* g~ceg

with cg as the conjugate of % in C. The mapping

Y--Y*

is an involutory ant i -automorphism of CG onto itself, i.e.

Y * * = ? , (7 -]- 8)* ---- Y* + ~*, (78)*----~'7" for all y ,~6CG.

Every S - r i n g T on G, and also 'every S-algebr~ Q T or C T is invariant

16 O. TAMASCHKE: On the Theory of Schur-Rings

under the mapping 7-~7". In particular, we have the equations

~ 1

In this section we merely give a fundamental definition which we shall use later. I ts usefulness is based on the following statement.

THEOREM 2.1 (WIELA~D~ [15], p. 386, foot-hole). - Let ~2 be a subfield of C

which is invariant under the mapping c ~ c (= the conjugate of c). Then ~ T is a semi-simple ~2-algebra.

PROOF. -- 12T is invariant under the mapping 7--'C* of CG onto itself. Therefore for every ~6~2T the equation :(~2T)z-----0 implies zz*(~':~)* ----- 0, hence ~*----0, and "~ ~ 0. It follows that ~2 T is semi-s imple .

In part icular, CT is a semi-simple C-algebra , and as such its representation theory is well known.

DEFINITION 2.2. -- Let D be a representation of CT qua C-algebra by matrices over the field C. The complex valued function

x : g ~ g x : = trace D~) for g6 ~ ,

on G is called the T-character o/ G related to D. The T-character X is called i rreducible i f D is irreducible.

For what is already known about T-character theory we refer to [3], [5], [6] and [7]. In this paper we concentrate on the elementary parts of S- r ing theory and on the information about the s t ructure of finite groups which can be obtained this way. In the next section we introduce the first concept that links S-r ing theory with group theory.

3 . - T-Subgroups.

A subgroup H of G is called a T-subgroup of G ([10], Definition 1.3) if

H ---- U : g , ,

i.e. if H is a union of T-classes. The meaning of the T-subgroups for the S - r i n g T on G is shown by the following.

TltEOREM 3.1. -- Let H be a subgroup of G. The following statements are equivalent.

O. TAMASCHKE: Or/ the Theory o/Schur-Rings 17

(1) H is a T-subgroup of G.

(2) H ~ T.

(3) ~<= H and H ~ H , ~ 6 ~ , are the S-classes of on S - r i n g on G contained in T.

(4) T N G / / H is an S - r i n g on G, and T A H / / 1 is an S - r i n g on H.

Note that, in an obvious notation, under the above conditions

S

t T(SG//H---- Z Z . H ~ H

TNIt/ /1----- E Z . %

PROOF. -

I. The equivalence of (1) and (2) is trivial.

II. Assume that (2) holds. Then ~ ; ~ H. Suppose that

H ~ H N H C ~ t H ~

Take any gEI2Cg~HNH~tH. Then there exist u, v, w, w 6 H , g~ECg~, g j 6 ~ i such that

g .~- ugly ~ wgjx.

Therefore

gi = u-lwgix~v -~ 6 H ~ j H

But HCgiH is a union of T-c lasses~s ince H z i H 6 T. Therefore ~ H ~ i H ,

and hence H ~ i H c H~jH. Similarly H ~ i H ~ H ~ H , and we obtain

The set ® = IHCgiHl i -~ 1, ..., t} yields a decomposit ion of G into non-empty, subsets which satisfies 1.1(1) and (2), and also (3) because of

( H ~ H ) -1 = H-Icgv~H -1 = H ~ t , H 6 ~ for all i = 1, ..., t.

For the element

H'ciH --- Z %g 6 T geG

we have c o ~ 0 if and only if g 6 HCg~H. By [17], Proposit ion 22.2, we obtain


18 O. TAMASCHKE: Or/ the Theory o] Schur-Rings

HCg~H E T for all i ~ 1, ..., t and also

t (HCg~H)(HCgiH)--- ~ b,ik1:k ~ T for all i, j-~- l, ..., t.

These p roduc t s are e l emen t s of T with the p r o p e r t y tha t they are i n v a r i an t by lef t and r igh t mu l t i p l i ca t i on with ev e ry e l em en t h E H. T h e r e f o r e

b~ik --~ b,iz if H~kH : HC~zH,

and each of these p roduc t s is a l i nea r combina t ion of the (k = 1, ..., t). I t fol lows that

t

S = ,,v Z . H ~ H

e lements HCgkH

is an S - r i n g on G con ta ined in T with the H ~ i H (i : 1, ..., t) as the S-c lasses of G. We have p roved (3).

I I I . Assume that (3) holds. T h e n S s T N G / / H . Tak e any z E T A G / / H .

t

~ : : v z(:~ because -cET.

": ---- h~: = ~h for all h E H because ": E G//H. T h e r e f o r e

z~-----z i if H ~ H - : H ~ H ,

and it fol lows that zE S. H e n c e T N G / / H - - - - S is an S - r i n g on G. Moreover H - ~ H C~IH is an S -c l a s s of G, and the re fo re

H = U ~ .

Obvious ly those T-c las ses which are con t a ined in H are the R - c l a s s e s of an S - r i n g R on H, and R ~ T N H / / 1 . T h u s we have p roved (4).

IV. Assume tha t (4) holds. E v e r y S - r i n g on H conta ins H. T h e r e f o r e

HE T N H / / 1 ~ T, and (1) and (2) hold.

The fol lowing r e m a r k may somet imes be useful .

LEMMA 3.2. -- Let H be a T-subgroup of G. Then

~HT.i~H ~ H ¢~iH for all i = 1, ..., t.

P R O O F . - By T h e o r e m 3.1 T A G / / H is an S - r i n g on G, and the sets


H'i~iH (i.-~ 1, ..., t )are the TNG/ /H-c lasses of G. It is Hz~H E T(3G//H. Therefore H"qH is a l inear combination

class sums, and

HztH ~--- n~. HCgiH

easy to see that of the T N G / / H -

must hold with a positive integer n~. To calculate n~ we count the elements which occur on both sides of the above equation with their multiplicit ies and obtain

and hence

t H t = I

DEFINI~IO~ 3.3. -- A subgroup H of G is called a T-subgroup of G i f 3.1(1)-(4) hold.

Thus the terms T-subgroup and T-subgroup (with T the ~S-semi. group on G related to T) are synonymous. Therefore we can use all results on T-subgroups proved in [10].

THEOREM 3.4 -- Let H and K be T-subgroups of G. Then the intersection H N K and the subgroup < H , K > generated by H and K are T-sub. groups, i.e. the T-subgroups of G form a sublattice of the lattice of all sub. groups of G.

PROOF. - [10], Theorem 1.5.

4 . - T-bTormal Subgroups.

As in Section 3 we start with the relevant notion of S-semigroup theory. A subgroup K of G is called T-normal ([10]. Definition 1,9) if K is a T-sub. group such float

K ~ ~ K holds for all i ~ 1, ..., t.

This definition of T-normal i ty is not sufficient for T-:normality. At least we do not know yet whether it is sufficient or not. The proper concept of T-normal i ty is based on the following statement.

T~EO~E~ 4.1. -- Let S and T be S-r ings on G. Let $1 be the S-class of G containing the unit element of G (which implies that $1 is a subgroup of G). Then the following statements are equivalent.

(1) $1 is in the centre of T and S ~ Tf~G//$1.


(2) S is a two-sided ideal of T.

(3) S c T and every S-character is a T-character of G.

If these conditions are satisfied then

holds.

n S = (n T)$~ = $~(n T) = $~(n T)$1

1 PROOF. I. -- Assume that (i) holds, e" = ~ , S_~ is a central idempotent

of QT. Therefore (QT)e is a two-sided ideal of QT. But e is also the uni t element of QS, and hence

Q(TA G//$1) -= QS = (QS)~ c (QT)e = ~(QT)e c Q(TAG//$, ) .

It follows that Q S = Q(T(SG//S,) is a two-sided ideal of QT, and

T S T ~ ' Q ? ) ( Q S ) ( Q T ) = QS.

Since every element of TST is a l inear combination of the with rational integers as coefficients, and lies in QS, it must in S. Therefore S is a two-sided ideal of T.

II. Assume that +2) holds. Let 7~ be an irreducible (Definition 2.2). Then

1 = E zx(g-1)g

T - c l a s s s u m s

be contained

S-charac te r of G

is a primitive central idempotent of CS with z a positive rational integer ([6], p. 218 or [7], p. 351). ~ also is a primitive central idempotent of CT since CS is a two-sided ideal of the semi-simple C-algebra CT. Therefore X, as the coefficient function of a primitive central idempotent of CT, is an irreducible T-charac ter of G (again by [6], p. 218 or [7], p. 351). Every S-character of G is a l inear combination of irreducible S-charac ters because CS is a semi-s imple C-algebra. Therefore every S-charac te r is also a T-character .

III . Assume that (3) holds. Let E be an irreducible S-charac te r of G. Then, as in I~,

1

is a central idempoten~ of CS. Since X is a T-character , and since CT is


a semi-s imple C-a lgebra ,

X ~ ~' a~xp

is a l inear combination of certain irreducible T-charac ters Xp with integers ap as coefficients. Each element

1 % = ~ g~ zex~ (g-~)g

positive

is a primitive central idempotent of CT, and

is a linear combination of them. Since the idempotents He are pairwise ortho. gonal, we obtain

Z~? 2

which implies

and hence

Fur thermore

za--2 = 1, Z~

~ E Hp. o

Let F be a representat ion of CS such that X is the S-charac te r of G affor. ded by F, and let Fe be an irreducible representat ion of CT such that X~ is the T-character of G afforded by Fe. Then

X(1) ~ trace F(e$i ) ~ degree F, since e$~ is the unit element of CS.

X~(1)~ trace Fr,(~:l)= degree /q'l), since ~:l is the unit element of CT.

( Z aexp(1))~ = X(1)2 = d i m c (CS)H ~< Z d i m c (CT)Hp = ~ Xp(1) 2.

This inequali ty can hold only for X ~ X p , and hence X is an irreducible T-character of G, and ~ ~--- Hp is a primitive central idempotent of CT. Conse-


queutly (CT)>7 is a minimal two-sided ideal of C T , as (CS)~ 7 is a minimal two-sided ideal of CS. But

dimc (CS)~ . = X(1) ~ - ~ Xe(1) 2 = dimc (CT)~,

and S ~ T by assumption. Therefore

( CS)~ = ( CT)~

is a minimal two-sided ideal of CT. As a semi-simple C-algebra CS is the direct sum of its minimal two-sided ideals, and therefore CS is a two-sided ideal of CT, and its unit element e$~ is a central idempotent of CT. It fol-

lows that S~ is in the centre of T Fur thermore

and therefore

C S --- ( C T ) ~ : ----- C(Tn G/t$~),

~' = T n G / / $ , .

Hence (1) holds, and our theorem is proved.

Dut+I~rI~ohr 4.2. - Let S and T be S-rings o~ G. Then S is called T-normal, i f 4.1(t)-(3) hold.

Not every S-r ing S on G which is contained in an S-r ing T on G is T-normal. S need not even be equal to an S-ring T A G / / H iwhere H is a T-subgroup of G (Theorem 3.1(4)). This is shown by the following example.

Let (7 be any primitive but not doubly transitive finite permutat ion group. Let ~1 be the stabilizer of one letter, and take T~--G//Cgl. Then

t

t ;> 2. Let S be the S- r ing on G with the S-classes ~1 and t 2 ~ . Then there i ~ 2

exists 'no T-subgroup H of G such that S-~-T(3G/ / I t , since there is no subgroup properly between ¢~1 and G.

Therefore those S-rings S on G which have the property that S ~ TN N G / / H for a subgroup H of G are a proper subclass of the class of all S- rings on G contained in T. They are in one-to-one correspondence with the T-subgroups el G. The class of all T-aocmal S-r ings is a proper subclass of them. We look at those T-subgroups now which are relatecl to the T-normal S-rings.

TI:fEOREM 4.3. - Let K be a T-subgroup of G. The~ the following statements are equivalent.

(1) K is in the centre of T.


(2) T N G / / K is a two-sided ideal of T.

(3) Every T N G//K-characler is a T-character.

The proof follows from Theorem 4.1 for S = T N G / / K .

D ~ T I ~ I O ~ 4.4. - A subgroup K of G is called T-normal , i f 4.3(I)-(3) hold.

Note that every T-normal subgroup is a T-subgroup of G by 4.3(1). 5Tote also that ZG-norma l i t y is normali ty in the ordinary sense.

PROPOSI~IO~T 4.5. -- Let K be a subgroup of G. Then the following stale- ments are equivalent.

(1) K is a T-normal subgroup of G.

(2) T N G / / K is a T-normal S-r ing on G.

We state relationships between various notions of normality.

PnoPom~IO~S 4.6. - Let S and T be S-rings on G, and assume S ~ T. Then every T-normal S-subgroup of G is S-normal. In particular every G / / ~ - normal T-subgroup, and also every normal T-subgroup, is T-normal.

The proof is obvious from Theorem 4.3(1).

THEORE~ 4.7. -- Let H be a T-subgroup of G, and K a T-normal sub. group of G. Then

(1) H K = KH is a T-subgroup of G.

(2) H K is a T n G / / H - n o r m a l subgroup of G.

4.7(2) is due to ALA~ CAMI~A.

PROOF. --

I. Since H E T , and K is in the centre of T, we o b t a i n H K - - - K H E T ,

hence H K = KIt. Moreover HK --~- I H N K t" HK. Therefore HKE T ([17], Propo-

sition 22.1), and H K is a T-subgroup of G.

II. For every • E T N G / / H we have

H ~ - - I H ] . ~ - ~ x H because ~EG//H,

Kx = : K because : E T and because of the T-normal i ty of K.

Iu QT the equation

1 HK = - - . H K IHNKI

24 O. TAMASCHKE: On the Theory o/ Schur-Rings

and the following computat ion hold.

( ) (1 1 . H K z ~ H K 5 = t H S K I _ _ I H ~ K I .H_ ~- ~HK.

Therefore H K is in the centre of T A G / / H , and hence H K is T n G / / H - normal.

THEOREM 4.8. - Let H and K be T-normal subgroups of G. Then HK is a T-normal subgroup of G.

1 PROOF. - HK is a subgroup of G by Theorem 4.7(1), and H K ~ ~ t . H K

is in the centre of T s i n c e H a n d K a r e in the centre of T.

THEOREH 4.9. - Let H be a T-normal subgroup, and K a subgroup of G containing H. Then K is T-normal i f and only i f K is T(~ G//H-normal.

PI~OOF. -

I. Let K be T-normal. Then K is TN G//H-normal by Theorem 4.7(%

II. Let K be TN G//H-normal. Using the T-normal i ty of H a n d wor- king in Q T we obtain for all :E T

1 , , ) .KIH HI= .IH HIK= Therefore K is in the centre of T~ and hence K is T-normal .

We generalize the notion of simplicity of groups to S-r ings .

DEFINITION 4.10. -- Let T be an S-ring on G different from the trivial S-ring G//G----{z . G I z E Z }. Then T is called simple, and G is called

lUsimple, i f there does not exist any T-normal subgroup differe~t from ~ and G.

The Homomorphism Theorem 6.2 will show that 4.10 is the correct definition of simplicity in the sense that T is simple if and only if the S-algebra ~2T has no homomorphisms other than monomorphisms and zero homo. morphisms.

We point to a special case of simplicity of S - r ings which has already appeared in the literature. According to WIELAND~ ([17], Definition 23.2) an S - r i n g T ~ G/ /G is called primitive, if there does not exist any T-subgroup different from % and G. Every primitive S - r i n g is simple, but there are simple S - r ings which are not primitive (take T ~ Z G for any non-abe. lian finite simple group G).

O. TAMASCHKE: Or/ the Theory of Schur-Rir/gs 25

We star ted this section by quot ing the corresponding notion of T-nor - mali ty in S - s e m ig roup theory, but so far we have not referred to it again. We will do so now. We repeat that T denotes the S - s emig ro u p on G related to T (el. Theorem t.15).

LWMMA 4.11.- Every T-normal subgroup of G is T-normal.

PROBLEM 4.12. - lS every T-normal subgroup K of G also T-normal in 61, i.e. does

K E T

always imply that

and K ~ t ~ i K for all i ~ 1 , ..., t

K'c, -~ ":tK for all i --- 1, ..., t ?

We can solve this problem for a class of S-r ings, namely for the double coset S-r ings . We prove a more general theorem, and, for lucidi ty 's sake, we repeat some s ta tements for double coset S- r ings which are already proved for arbi t rary S-r ings . Refer also to [8], Theorem 2.3 and Defini t ion 3.1. We give 6 equivalent s tatements , two purely group theoretical, two r ing theoretical, and two character theoretical.

IH G denotes the (ordinary) character of G induced by the 1-charac te r 1~ of the subgroup H of G, and similarly with 1K ~ for a subgroup K of G in- stead of h r. As usual (.,.)G means the inner product of characters.

TttEOREYI 4.13. -- Let H and K be subgroups of G. Then the following statements are equivalent.

(1) H <-- K and HgK---- KgH for all gE G.

(2) K--- H(KV~K ~) for all gEG.

(3) K is in the~eentre of the double coset S-r ing G/ /H.

(4) The double coset S-ring G / / K is a two.sided ideal of G/ /H.

(5) Every G//K-character is a G//H-eharacter of G.

(6) H <- K and (1H ~, 1 ~ ) ~ = ( l ~ , IK~)~.

The equivalence of (1) and (6) was overlooked by the author and was pointed out to him by J. L. ALr'ERI~ dur ing the Algebra Sympos ium 1966/67 at the Univers i ty of Warwick, Coventry, England.

PROOF. -- The equivalence of (t) and (2) holds for arbi t rary groups and was proved in [10], Theorem 1.15.

Under the addit ional assumpt ion H_- < K the equivalence of (3), (4),



(5) is a special case of Theorem 4.3 for T ~ G//H. Also under the additional assumption H - < K the equivalence of (3), (4), (5), (6) has been proved in [8], Theorem 2.3. Yet the assumption H - - < K is superfluous. (3) and (4) certainly imply H ~ K. That (5) also implies H = < K can be seen as follows.

Our proof 4.1, I I I that 4.1(3) implies 4.1(l) used the assumption S~T at only two points. Without that assumption we have shown that every primitive centrat idempotent of CS is also a central idempotent of CT. Since the unit e lement aS1 of CS is the sum of all primitive central idempo teats of CS (by the semi-simplicity of CS), e$, is a central idempotent of CT. If we apply that remark to S---- G / / K (where ~g is the unit element of CG//K) and to T-----G//H, it follows that K is in the centre of G//H,

hence H = < K and G / / K ~ G//H. Therefore the equivalence of (3), (4), (5), (6) holds.

(1) implies

(I') H g K = K g K for all gCG.

Conversely, (1') implies

H ~ HK ~ HIKe-- KIK--.-~- K and

Hgg = Kgg = (Kg-lg ) -1 = (Hg-Ig) -1 = KgH for all g~-G.

Hence (I) and (1') are equivalent.

It is well known that (1~, lEG)+ is equal to the number of double eosets

HgK, g E G, and (1g G, 1KG)O is equal to the number of double cosets KgK, g E G. Therefore (1') and (6) are equivalent, and our theorem is proved.

The double eoset S-semigroup G/H is the S-semigroup related by Theo- rem 1.15 to the double coset S - r i n g G//H. A subgroup K of G is a G/H- subgroup if and only if H-- < K. The G/H-classes of G are the double cosets [IgH, g E G. Hence a subgroup K of G is G/It-normal if and only if

H<=K and K(HgH)-~(HgH)K for all gEG.

But H ~ K implies H g K ~ (HgH)K and K g H ~ K(HgH) Therefore as a consequence of Theorem 4.13 we obtain

THEORE~I 4.14. - Let K be a subgroup of G. Then K is G//H-narmal if and only i f K is G/H-normal.

Thus the question 4.12 is answered affirmatively for double eoset S-r ings. Therefore all statements on G/H-normal subgroups automatical ly hold for finite groups with G//H-normality as well. We refer the reader to [10], 1 .16- 1.21.

O. TAMASCHKE: Or/ the Theory o/Schur-Rir/gs 27

5. - Homomorphisms .

To the notations introduced at the beginning of Section 2 following.

F is

S is

E is

we add the

a finite group.

an S- r ing on /7'.

the S-semigroup on F related to S (Theorem 1.15).

®----{$1, ,.., $~ } is the set of all S-classes ( ~ E-classes) of F.

s t - " t Sj] the number of elements in the j - th S-class of F.

a1 ~ $i ( j ~ 1, ..., s) the S -c l a s s sums of F.

is a subfield of C which is invariant under conjugacy c---c.

~0 is a homomorphism of the S -a lgebra ~2T on G into the S -a lgeb ra ~2S on F.

¢ is the homomorphism related to ~ of the S - semig roup T on G related to T into the S-semigroup ~ on F related to S (Theorem 1.16).

T is the set of all non-empty subsets of G which are unions of T-c lasses ( ~ T-classes).

is the set of all non-empty subsets of F which are unions of ~,-classes ( = S-classes).

can be uniquely extended to a mapping (I) of T into Z ([10], Proposi- tion 2.2) by

X ~ : ~ L) ~¢ for all X E T .

Let H be a T-subgroup of G. Note that H need not be an element of

T, but we have H E T--. By [10], Proposi t ion 2.4(1), H ~ is a E-subgroup, and hence it is an S - subgroup of /7.

P R O P O S I T I O N 5 .1 . --

i.e. ~ maps every subgroup average of a average of an S-subgroup of F.

PROOF. - Because of HE T we have

T-subgroup of G onto a subgroup

1 1 ~. ~ ~ t ~ ,

28 O. TAMASCHKE: OF/the Theory o] Schur-Rings

From 1

/ 4 ~ ~ = (~ /~ = ~ and ,~,~ = =~:.~;~(~.1~) ~x,;l

we obtain H*-/-/~ ~ H ~, and hence (by the finiteness of F) we have another

proof of the fact that H ~ is an S-subgroup of F. For every T-class average % such that " ~ i ~ H the equation

holds. The analogue holds for the S-class averages of F contained in //¢ which, by Definition 1.5(3) and 1.16, implies

Therefore

On the other hand

¢ ~ ' ~ X ~ a n ~ H ~ for all ~ , c H .

8j~R ¢

For every $ i ~ He there exists ~ ~ H such that

Hence we obtain

and Proposition 5.t is proved. Moreover the proof yields some numerical relations which we deduce

from the equation

1 21 C , = ~ = ~ . = 1 ~ t ,~ , ,

and which we state as

PROPOm~ION 5.2. - ]Hl.s j = IH~I. Y t, for all 8i ~ t t ~.


CO~tOLL~a¥ 5.3. - I f ~ is an automorphism of ~2T, then

~ = ¢61 implies t~ = t t.

With every T-subgroup H of G there are associated (Theorem 3.1(4))

the S- r ing T A G / / H on G,

the S- r ing T A H / / 1 on H.

Let us study the effect of ~ on these S-rings.

PROPOSITZO~ 5.4. - kS N GTP/ / H ~ is an S-ri~g on G ~, and the restriction of ~. to ~2(TN G/ /H) yields an epimorphism of the S-algebra ~2(TA G//H) on

G onto the S-algebra g/(SN G~//H% on G ~.

PRooF. - By Theorem 3.1(4) S N e l l 1 is an S - r ing on G ~ since G ~ is

an S - s u b g r o u p of F. A subgroup of G ~ is an S - subg roup of F if and only

if it is an S n G ~ / / 1 - s u b g r o u p of G ~. Therefore S N G X / / H ¢ = ( S A G ~ / / 1 ) A

N G¢/ / t t ~ is an S - r i n g on G ~. The images of the TN G//H-class averages (Lemma 3.2) are

which are S n G$//H~-elass averages, again by Lemma 3.2. Since

holds, the restrict ion of ~ to ~2(TAG//H) yields an epimorphism of the

S -a lgeb ra 12(TNG//H) on G onto the S-a lgebra ~ 2 ( S N ~ / / H ~) on G ~.

PRoPosI~zo~ 5.5. - S A H ~ / / 1 is an S-ring on I t ~, and the restriction of ¢p to ~ ( T A H / / 1 ) yields an epimorphism of the S-algebra ~2(TNH//1) on

H onto the S-algebra ~2(SAH~//1) on t~ .

The proof is obvious.

COROLLARY 5.6. --

(~2 T)~-- Im ¢? = ~2(SN 6®//1)

is an S-algebra on G ~, i.e. every homomorphic image of an S-algebra is an S-algebra.


Paot , osITIo~ 5.7. - Let K be a T-normal subgroup of G. Then K ~ is an

SN G~ / /1 -normal subgroup of G ~.

PRoof. - - S g is in the centre of ~ T. Therefore ~g ~ ~ is in the centre

of Im ~ ~ ~2(SNG6//1), and hence K¢ is an (S(~ G$//1)-normal subgroup of G¢.

Let L be an S-subgroup of F contained in G% Then the inverse image

L ~ - ~ : ~ U ~

is a T-subgroup of G ([10], Proposition 2.5(1)), and hence it is a T-subgroup

of G. )[oreover (L+-~) + = L holds ([10], Proposition 2.5(2)).

We continue our discussions of homomorphisms, but we need a new concept, and therefore we start a new section.

6. - The Homomorphism Theorem.

If a is a homomorphism of the group G, then K e r a is a normal subgroup of G, the cosets gKer ~, g E G, are the sets of those elements of G which have the same images under :¢, and lm ~ is isomorphic to the factor group G/Ker a. 0 u r next results can be considered as a generalization of these group theoretical facts for finite groups, which we obtain from specializing T = Z G . We keep all notations introduced earlier in this paper.

D]~:~I~:[~Io=~ 6.1. - Let ~ be a homomorphism of the S-algebra 12T on into the S-algebra ~2S on F. Then

K e r ~ : = U ~

(where'1 is the unit element of ilS) is called the kernel of %

S ~-1 where $1 is the S-class of /7 'conta ining the unit Note that Ker ~ ~ i ,

element of F. ]~Ioreover $1 is an S-subgroup of F contained in G ~, and therefore Ker %o is a T-subgroup of G. So the kernel of an S-a lgebra homomorphism of ~2T is not a substructure of the S-a lgebra ~2T, but a subgroup of G.

THE HOMOMORPHISM THEORE~ 6.2. -- Let ~ be a homomorphism of the S-algebra ~2T on G into the S-algebra ~ S on F, and set K ~-Ker ~. Then

(1) 7:=~----~ i f and only i f ~ , K = ~ jK .

(2) K is a T-normal subgroup of G, and henoe the S--ring TN G / / K on


O ~vhich has the sets ¢g~K-~ K ~ K ( i= 1,...,t) as the (TNG/ /K) - classes of G) is a two.sided ideal of T.

(3) The restriction of ~ ~o ~2(T(~ G//K) yields an isomo~Thism of the S-algebra ~2(T~G//K) on G onto the S-algebra Im ~0-----~2(S~

G~//1) on G ~.

Pl~oo~' I. - K is a T-subgroup of G, and ~ ---~ ~ holds by Proposition 5.1 and Definition 6.1.

Assume that " ¢ ~ - ~ . Then~ using ( ~ ) * = ~ i ) ~ (Definition 1.5(3)), o"1 ap- pears in

with a positive coefficient. Therefore there exists ¢g~cgTwg~ such that

~'~¢ ~--- (~, that is ~l ~ K. Take any g E ¢gz. Then there exist x E ~ i and y E ~ i such that g---y-~x, hence

x --- yg E ¢gjcG~ £ ¢gjK.

But ~ j K is a union of T-classes, because ~jKET. Therefore ~ ~iK,

and hence ~ K ~ j K . Interchanging i and j we also and so

~ ~ ~ implies ¢ g i K = ~ i K .

Since

and therefore

obtain ¢giKc: ~K~

~ k ~ ~ iK implies ~-----~'],

: ~ z ~ if and only if ~iK~C~iK.

Thus we have proved (1). Another proof of (1) can be obtained from Theorem 1.!6 and [10], Theorem 2.7. Furthermore, from the foregoing it is clear that

= if and o n l y if

Thus, using Lemma 3.2, (3) holds as well.

II. Let J be the set of all z E T such that -c~ ~ 0, that is the kernel of ~ qua algebra homomorphism. Each T-class average

3~ O. TAMASCHKE: Or/ the Theory o] Schur-Rings

can be decomposed into the (T~ G/ /K) -c las s ave rage zKzi~K (Lemma 3.2) and

the e l emen t z~--~K'~i$K of J . T h e r e f o r e

and f rom (3) we obta in

I f the e l emen t

T = (TN G/ /K) + J,

(TC~ G / / K ) O J = O.

t ~ t $

t=1 i = l i = l x~ E gt,

of T is con ta ined in J, then

t

since ~ i nduces an i somorph i sm of 7 ' c ~ G / / K onto Im ~. T h e r e f o r e J is gene ra ted , as an ~2-vector space, by the e l emen t s

% - eK%eK (i = 1, ..., t).

L e t -t~ be the uni t e l emen t of the two-sided ideal J of the semi-s imple ~2-a lgebra ~2T (Theorem 2.1). Th en ~ is a cen t ra l i dempo ten t of 12T, and we have also

t

T h e r e f o r e

SK~ ~ K ~ ~ - - ~ ~K~K~- - -O ,

and, s ince e ~ - ~K ~ £~1 - - ~K~i~K E J, this means that

I t follows that e K = e ~ i - - ~ is in the cen t re of ~ T , and the re fo re K is

in the cen t r e of T. W e have p roved that K is a T - n o r m a l subgroup of G, and the re fo re ~iK- - - K ~ i - - - K ~ i K for all i---- 1, ..., t.

THE CAI~ONI0 EPII~ORPI-IIS~ THEOREI~ 6.3. -- Let K be a :l~normal sub. group of G. Then

(I) ?g:z---~zsK is an epimorphism of the S-algebra ~2T on G onto the T-normal S-algebra ~ ( T A G / / K ) on G.


(2) Ker ~0K -~ K.

We call q~K the canonic epimorphism or the projection of f~T onto ~2(TfqG//K), and we consider the T-normal S - r i n g T N G / / K as the factor S-ring of T modulo K. In order to keep in accordance with the notation in group theory, we write T/ (T~K/ /1 ) : ~ T fqG/ /K for the factor S - r i n g of T modulo K.

PROOF. - eg is a central idempotent of ~2T. Therefore ~g is an epimor. phism of the ~2-algebra ~2T onto the E~-algebra ~(T(qG//K). By Lemma 3.2 the image

= =

of any T-c lass average is a (TfqG//K)-class average. Hence 1.5(3) holds for q~g, and q~c is a homomorphism in the sense of Definit ion 1.5. Our sta-

tement (2) follows from the fact that ~ is the ~ of Im ~K---12(T~G//K).

Theorems 6.2 and 6.3 show that the T-no real subgroups of G are exactly the kernels of the homomorphisms of the S-~ gebra ~2T.

We draw a couple of conclusions from the Homomorphism Theorem 6.2. We start with rephrasing Proposi t ion 5.2.

THEOREM 6.4. -- Let t l be a T-subgroup of (~, and ~ a, homomorphism of the S-algebra ~2T on G into the S-algebra i~S on F. Then

IHI.1SJI=IH~I.I~,(HAKer¢~)] for any ~,c=H such that ~ = $ j .

P R O O f . - We set K = K e r ~ and assume that ~ - - - - ~ . Then ¢g~K----- ~iK (Theorem 6.2(1)) and for every giE ~ there exist g j E ~ j and k E K

such that g~=gjk. If we further assume that ~ , ~ j ~ H , then kEH(3K.

Therefore ~ i ( H N K ) . Conversely, ~k~cgi(H(qK) implies ~ - - - - - ~ . It follows that ~i(H(qK) is the union of all those T-classes ~ H for which

¢g~ ~ ~ ' holds, and our theorem follows from Proposit ion 5.2.

As a special case of Theorem 6.4 we take the S-c lass $1 of F which

contains the unit element of F. Then S~ is a subgroup of F contained in H ~ For the T-c lass "~t of G which contains the unit element of G, we have

1 = $1 and ~ 1 ~ H f q K e r ~ . Thus we have proved

Tlt:EORE~ 6.5. - Let H be a T-subgroup of G, and ¢~ a homomorphism of the S-algebra ~2T on G into the S-algebra ~2S on F. Then

IH : H A K e r q~l ~-~ IH~:S~I •


34 O. TAMASCHKE: Or/ the Theory o/ Schur-Rings

In particular

1G:Ker ~l = IG~:$~I •

THEOREM 6.6. -- Let ¢~ be a ho,momorphism of the S-algebra ~2T on G into

the S-algebra ~2S on F, and L an ( s n ~ / / 1 ) - n o r m a l subgroup of G~. Then

L ~-~ is a T-normal subgroup of G.

PI~OOF. - By the (S~G~/ /1 ) -normal i t y of L we have t 2 (SNG~/ /L) as

a two-sided ideal of I m ~ = ~2(SN G g / / 1 ) . The set J of all • E f i t such that

zgEgt(sn G~//L) is a two-sided ideal of gtT. We set K = Ker ~. The two.sided ideal of those elements "~ E ~T such that ":~ = 0 (that is the kernel of ¢p qua algebra homomorphism) is equal to (t2T)(ag-- eK)(cf, part I I of the proof

of 6.2) and is contained in d. Let ~ be the uni t element of J. We set H =

= L g-*. Then we have also e~ = aL. Therefore ~ - - ~H E (~2 T)(e% -- eg). Cer-

tainly ~ =<= K~_ H holds since ~ = $~ =c L for all '~,,.c ...... K. Hence

Since all of ~, e%, ~: are in the centre of AT, so is ell, and therefore H is T-normal .

As in ordinary group theory the correspondence between T-subgroups

of G and the S-subgroups in G $ becomes a one-to-one correspondence if we restrict ourselves to those T-subgroups of G which contain the T-normal subgroup Ker ~. We denote by < X , Y ~ the subgroup of G generated by the subgroups X and Y of G.

T~EORE~I 6.7. -- Let ~ be a homomorphism of the S-algebra ~2T on G into the S-algebra gtS o7~ F. Let X and Y be T-subgroups of G containing Ker% Then

(1) (X¢) ~-~ = X.

(2) ( X N Y ) ¢ = X ~ N Y g and <X,Y>~---- -<X%Y%>.

(3) X is a T-normal subgroup of G i f and only i f X ~ is an (SNG¢//1) -

normal subgroup of G ~.

Therefore, taking (L~- I )*= L for all S-subgroups L <- ~ into account,

([10], Proposition 2.5(2)), the mapping X--~ X ~ is an isomorphism of the lattice of all T-subgroups of G containing Ker ~ onto the lattice of all S-subgroups

O. TAMASCHKE: Or/ the Theory o] Schur-Rings 35

of G ~, which brings into one-to.one correspondence the T-normal subgroups of

G containing Ker q~ wi[h the (SAG$/ /1 ) -normal subgroups of G ~.

I~ROOF. -- (1) and (2) merely restate [10], Theorem 2.10, because of Theo- rem 1.16 and Definition 3.3. Statement (3) follows from Proposit ion 5.7, Theo-

rem 6.6, s tatement (1), and from (L;~-~) ~ - - L for all S - subgroups L < G ~.

A homomorphism of a group is a monomorphism if and only if its ker. nel is the unit subgroup. The analogue holds for S -a lgebra homomorphisms as is shown in the following.

PRoPosit ion- 6.8. - Let "~ be a homomorphism of the S-algebra ~2T on G into the S-algebra ~2 S on F. Then ~ is a monomorphism i f and only i f Ker ~ = ~ .

PROOF. - If %o is a monomorphism, then v~ is the only T-class average

mapped onto z~, hence Ker~- -~c~ ,

If Ker ~ - - ~ , then ~,~ # z~ for i ~ j because of Theorem 6.2 (1). Since

the ~ ( i = 1, ..., t) are all of the Im ~ = ( S ( 3 G ~ / / l ) - c l a s s averages of G $, they are an ~2-basis of Im % and hence ~ is a monomorphism.

We have defined the concept of S-a lgebra monomorphism in a na ive way as an injective homomorphism (Section 1 after Definition 1.5). In the category of all S-a lgebras over ~2 a homomorphism ~ of an S-a lgebra F~T on G into an S-a lgebra ~2S on F is called monie, if [or every pair ~1, ~z of homomorphisms of any S-a lgebra ~2R on any finite group E into ~T the equation

~ ~ ~.~ implies ~ ~- ~ .

As an application of the foregoing results we prove

THEOREM 6 . 9 . - Let ~ be a homomorphism of the S-algebra ~T on (~ into the S-algebra ~2S on F. Then ~ is a monomorphism i f and only i f ~ is monic in the category of all S-algebras over ~2.

PRooF. - I. Let q~ be a monomorphism. Then the injectivity of ~ as a mapping immediately yields that ~ is mosaic.'

II. Assume that ~ is monic. Set K : Kerg . Then, since K is a T-sub . group, R ~ T A K / / I is an S - r ing on K contained in T (Theorem 3.1(4) and Definition 3.3). Let ~ be the injection of ~/~ into ~T , and 4.2 the zero homomorphism of ~2R into ~2T (cf. the proof of Theorem 1.6). Then ~ ~---~0 holds since ~0 induces the zero homomorphism of ~2R into ~2S. It follows that ~ :--~z because ~ is monic. Therefore K has only one R-c lass , n~mely ~ , . Hence K ~ ~ , and ~ is a monomorphism by Proposi t ion 6.8.

36 O, TAMASCHKE: On the Theory of Schur-Rings

7. - T h e I s o m o r p h i s m T h e o r e m s .

We want to emphasize that S - r ing theory can be considered as a generalization of finite group tt::eory. In this context the Isomorphism Theorems play an important r01e. They have already been stated for double coset S-r ings in [8], Theorems 3.i0 and 3.11.

TI=[E FIRST ISO~IORP]t:IS3~ TttEOREM 7.1 . - Let K be a T-normal subgroup of G, and L a T-subgroup of G containing K. Then

(1)

(2) (a)

(b)

(c)

L is T-normal if and only if L is (TN G//K)-normal.

Assume that L is T-normal. Then

T / ( T A K / / I ) : ~ T N G / / K is the factor S-ring of the S-ring T on G modulo the T---normal subgroup K.

(TN G//K)/((TN G//K) N L / / 1 ) : : (T n G/ /K)N G/ /L = T n G//L is the factor S-ring of the S-ring TN G/ /K on G modulo the (T(h G / /K) - normal subgroup L.

T/(TNL//1): ~ TNG//L is the factor S-ring of the S-ring T on G modulo the T-normal subgroup L of G.

The iterated factor S-ring (b) (qua factor S-ring of the factor S-ring (a)) is equal to the factor S-ring (c).

7.1(1) repeats Theorem 4.9. The isomorphism statement of 7.I(2) is even an equality, and a tautology. The reason for this peculiari ty is the fact that for S- r ings the factor s t ructure also is a substructure.

TKE SECOND ISOMORPHISM THEORE~ 7.2 . -- Let tl be a T-subgroup of ~, and K a T-normal subgroup of G. Then

(1) H A K is a (TNH//l)-normal subgroup of H.

(2) K is a (TnHK// l)-normal subgroup of HK.

(3) The S-algebra ~2(TN(H//HNK)) on H is isomorphic to the S-al. gebra ~2(TAHK// K) on HI(.

Note that TN(H/ /HNK) is the factor S - r ing of the S - r i n g T N H / / t on H modulo the (TAH//1)-normal subgroup HNK, and T N H K / / K is the factor S - r i n g of the S - r i ng TNHK/ /1 on HK modulo the (TNHK//1)-normal subgroup K.

PBoo~ ~. - Take the projection ~K of the S-a lgebra ~T on G onto the factor S-a lgebra ~2(T(h G//K) on G (Theorem 6.3(1)). ?g induces an epimorphism

of the S-a lgebra ~2(TNtt//1) on H onto the S-a lgebra ~2(TAHK//K) on


HK. Then HNK~-- -Ker ~ is a (TNH//1)-normal subgroup of H (Theorem 6.2(2)), and the S -a lgebra t2(TV~(H//HNK)) on H is isomorphic to the S-algebra I m v ~ t 2 ( T V ~ H K / / K ) on HK (Theorem 62(3)). K is in the centre of

T, since K is a T-normal subgroup of G. Therefore K is also in the centre

of T N H K / / 1 , and hence K is a (TV~HK//1)-normal subgroup of ItK.

All group algebras of finite groups over ~2 are 6bjeets of the category o[ all S-a lgebras over £t, and all group homomorphisms (qua homomorphisms of the related group algebras) are morphisms of that category. Yet the group algebras and the epimorphism.+ of S-a lgebras onto group algebras are era. bedded in the category of all S- :dgebras over 12 in a relnarkable way.

TttEORE:M: 7.3. -- Let ~ be an epimorphism of the S-algebra ~2T on G onto the group algebra gtF of F (qua S-algebra on F). Then

(1) Ker ~ is a normal subgroup of G.

(2) The group G / K e r ~ is isomorphic to the group F.

(3) :¢:g----+~ for g E ' ~ , is art epimo~Thism of the group G onto group F, and Kcr ~ = Ker q~ holds.

PROOF. - For any x,yC G there exis~ i, j, n E{1, ..., t} such that

the

Then

zcE~; , y e l l , x y E ' ~ , , .

t

is a single element f ~ F (Definition 15(3)), which implies

= ~ = f for all k such that a~tk:>0, and t

Since a;i~ ~ 0, it follows that f - - - -~ = (xy) ~ and therefore

(wy)~ = x ~ y ~ for all ,x,yE G.

is an epimorphism of the group G onto the group F. For g E ~ we have

g ~ - ~ l if and only if ~ ; ~ K e r %

Therefore Ker ~ ~ Ker ~ is a normal subgroup of G. Oar theorem is proved.


8. - T - S u b n o r m a l S u b g r o u p s .

DEFI~I¢IO~ 8.1. -- A subgroup L of G is called a T-subnormal subgroup of G, i f there exists a chain

G = Lo > L~ = :>L,.---L_

of T-s~tbgroups of G such that L~ is a (T(~Li_~/ /1)-normal subgroup of L~_~ for all i ~ l, ..., r. Such a chain of subgroups is called a T-subnormal chain of G, and the S-r ings TAL~_~//L~ ( i = l, . . , r) are called its factors. Any T-subnormal chain from G to the T-normal subgroup ¢g~ is called a T-subnormal series of G.

For the S-semigroup T related to the S - r ing T (Theorem 1.15) the notion of T-subnormal subgroup can be introduced by substi tut ing T-normal i ty in Definition 8.1 ([11], Definit ion 1.1.) A theory of T- subnormal subgroups has been initiated in [11]. We shall note the differences between T-subnor- mality and T-subnormal i ty .

PnoPosI¢Io~ 8.2. - Every T-subnormal subgroup of G is T-subnormal.

PROOF. - Lemma 4.11.

Whenever T-normal i ty coincides with T-normal i ty , then the T-subnormal and the T - subnorma l subgroups are one and the same. For instance, a subgroup K of G is G / / H - s u b n o r m a l if and only if it is G/ t t - subnormat (Theorem 4.14). Yet, even if Problem 4.12 were answered in the affirmative, that is if T-subnormal and T-subnormal subgroups were always the same, there would still remain a difference between T-subnormal chains and T- subnorma l chains, because the factors of a T-subnormal chain are S-r ings whereas the factors of a T - subnorma l chain are S=semigroups ([ill, Defini- tion 1.l). Therefore the theory of T-subnormal chains is not made superf luous by the theory of T - subnorma l chains.

Even if Problem 1.19 could be answered in the affirmative, that is if every S-semigroup on a finite group were related to an S- r ing in the sense of Theorem 1.15, this would not be sufficient to deduce the theory of T-normal i ty from that of T-normal i ty nor vice versa, i f an even stronger statement than the affirmative answer of Problem 1.19 were true, namely that not only is every S-semigroup on a finite group related to an S - r ing in the sense of Theorem 1.15, but also that every S-semigroup homomorphism is related to an S -a lgeb ra homomorphism in the sense of Theorem 1.16, then the theory of T-subnormal i ty could be deduced from the theory of T-subnormal i ty by the functor of Theorem 1.i7. And only if the functor of Theorem 1.17 had an inverse could the theory of T-subnor- mali ty be deduced from the theory of T-subnormal i ty .


These considerations may jus t i fy the remainder of this paper which is analogously writ ten to [11].

LEMI~[A_ 8.3. -- Let K be a T-subnormal subgroup of G, and let L be a ( T N K / / 1 ) - s u b n o r m a l subgroup of IV. Then L is a T-subnormal subgroup of G.

PnooF. - Let .K -~ Lo ->_- L~ >= ... ~ L,. ~ L be a ( T N K / / l ) - s u b n o r m a l chain of K. Then L~ is a ( ( T N K / / 1 ) n L i _ J / l ) - n o r m a l subgroup of L~_~. Since ( T N K / / 1 ) A L i _ , / / i ~ T N L t _ ~ / / 1 , this chain extends every T-sub. normal chain from G to K to a T-subnormal chain from G to L.

Tm~ORI¢~ 8.4. -- Let H be a T-subgroup of G, and let L be a T-subnor. real subgroup of G. Then H A L is a (TNH/ /1) -subnormal subgroup of H.

PROOF.- Take any T-subnormal chain from G to L as in Definition 8.t. H ~ H N G : H N L o is ( T N H / / i ) - n o r m a l , and hence ( T N H / / 1 ) - s u b . normal. Assume that HNL~_~ is already shown to be (T(YH//1)-subnormal . Since Li is a (TNLi_~/ /1) -normal subgroup of L~_~ by assumption, the Se- cond Isomorphism Theorem 7.2 yields HALt- - - - (HNLt_~)NLi as a (TN(HN nL~_i) / /1) -normal subgroup of HNLi_~. Therefore H N L , is a ( T N H / / 1 ) - subnormal subgroup of H by Lemma 8.3. That proves our theorem.

THEOREM. 8.5. -- Let K and L be T-subnormalsubgroups of G. Then K N L is a T-subnormal subgroup of G.

PROOF. -- K N L is a ( T N K / / 1 ) - s u b n o r m a l subgroup of K by Theorem 8.4. Therefore K A L is a T-subnormal subgroup of G by Lemma 8.3.

Some fur ther statements on the special case ol G/ /H-subnormal (= G/H- subnormal) subgroups can be made. We refer the reader to [11], Theorem 1.5, Corollary 1.6, and Theorem 1.7.

9. - The Four Subgroup Theorem.

In the last section we have dealt exclusively with the T-subnormal subgroups. Now we turn to the factors of the l - s u b n o r m a l chains. Our aim is to prove a Jordan-I t i i lder Theorem for T-composition chains. As in ordinary group theory a basis for these investigations is Zassenhaus' Lemma whose S- r ing version we are now going to formulate and to prove. It is almost entirely dependent on the validity of the Isomorphism Theorems for S-algebras.

TI-IE FOUR SUBGROUP THEOREM 9.1. -- Let Ko, K, Lo, L be T-subgroups of G. Assume thai Ko is a ( T N K / / 1 ) - n o r m a t subgroup of K, and L o a ( T N L / / 1 ) - n o r m a l subgroup of L. Then


(l) (KNLo)Ko is a (TN(KNL)Ko//1)-normal subgroup of (KNL)Ko.

(2) (Kon L)Lo is a (TN (KN L)Lo//1)-normal subgroup of (KN L)Lo.

(3) The S-algebra

~t(TN(KNL)Ko//(KNLo)Ko) o~ (KNL)Ko

is isomorphic to the S-cdgebra

gt(TA(KNL)Lo//(KoGL)Lo) on (KGL)Lo,

(4) ! (KN L)Ko:(KQ Lo)ho ) = I(KG L)ko :(Ko N L)Lo !,

PROOF. - KQLo, / ( N L , Ko are ;/ '-subgroups of G contained in K (Theo- rem 3.4), Therefore they are (TNK//l)-subgroups of K as welt. Every (TN NK//1)-normal subgroup of K is permutable with every (TNK//1)-sub- grotlp of K (Theorem 4.7(1)). It follows that (KNLo)Ko and (KNLj'Ko are in- deed (TnK//1)-subgroups of K (Theorem 3.4). By the same arguments (KoN NL)Lo and (KNL)Lo are (TnL//1)-subgroups of L.

KoNL~(KNL)V]Ko is a (TN(KnL)//l)-normal subgroup of KNL, and ~:':--~'Cego is an isomorphism of the S-a lgebra t2(TN(KNL)//(KoNL)) on KNL onto the H-algebra ~(TA(KNL)Ko//Ko) on ~KNL)Ko by the Second Iso- morphism Theorem 7.2.

Also KALo is a (TN(KNL)//l)-normal subgroup of KNL, and hence (KoNL)(KNLo) is ~ (Tn(KnL)//1)-normal subgroup of KNL (Theorem 4.8). (KoNL)(KNLo) is also a (TN(K~L)//(KonL))-normal subgroup of KNL. Therefore ((KoN L:(K N Lo)) ~ --- (Ko n L](K N Lo)Ko ~ (K n Lo)Ko is a (TN (KN nL)Ko//Ko)-normal subgroup of (KNL)Ko (Propesilion 5.7). By the First Iso- morphism Theorem 7.l (KNLo)Ko is also a (Tn(KNL)Ko//1)-normal subgroup of (KNL)Ko, Thus we have proved (1), and (2) follows by symmetry.

The isomorphism ~ maps the (TN(KNL)//(KoNL))-normal H-algebra (i.e. the two sided ideal, (.f. Theorem 4.1 and Definition 4.2) gt(TN(KNL)//(KoA NL)(KN Lu)) isomorphically onto the (TN,K N L)Ko//Ko)-normal H-algebra ~2(TN(KNL)Ko//(KNLo)Ko). In the same w~y the H-algebra Vt(TN(KNL)// //(KoNL)(KNLo)) is proved to be isomorphic to the S-, 'dgebra ~2(TN(KN NL)Lo//(KoNL)Lo), and hence (3) holds. (4) follows from

t (K n L)Ko :(K N Lo)Ko ! ~- ) (K N L) :(Ko N L) (K N Lo) : = ! (K N L)Lo :(Ko n L)Lol

by an elementary computation. The first consequence of tile Four Stlbgroup Theorem for the investiga-

tion of the factors of T-subnormal chains is the following.


T H E R E F I N E M E N T THEOREM FOR T-SUBNORMAL CHAINS 9 . 2 . - A s s u m e that

( i) G ~- Ko = > K1 = > ... = > K,, = L

( i i ) G ~ L o > L I = > . . . = > L s = L

are t w o T - s u b n o r m a l chains f rom G to the T - s u b n o r m a l subgroup L. Set

K~,j ~ Ki(Ki-1N Li) and Li,~ -~-- Li(Li_~ n K~)

for i = 1, ..., r and ] = 1, ..., s. Then

(1) Ki-1 K > K;, > = ~ , o = 1 . . . . > K i , s = K ~

is a (TN K ~ _ I / / 1 ) - s ~ b n o r m a l chain f rom K~_ ~ to K~ for each i : 1, ..., r.

(2) Li_~ = Li, o > Li,~ >=... > Li, r --~ L i

is a ( T n L j _ , / / 1 ) - s u b n o r m a l chain f rom Li_t to L j f o r each j = 1, ..., s.

(3) The S-algebras

~2(TnK~, i_~/ /Ki , i) a n d ~ ( T N L i , i_x / /L i ,~ )

are i somorphic : for all i --~ 1, ..., r a n d all j----- 1, .... s.

(4) I K~, i - , :gi, i] = l Li, i-x :L],i l

for all i = 1, ..., r a n d all j = 1, . . . , s.

Jo in ing the chains (1),~respectively (2), together, we obtain ref inements o f the T - s u b n o r m a l chains (i) a n d (ii) for which

T n K~, i-~ I/K~, j ~--> T n L j, ~_~ I I Li,

is a one.to.one correspondence of their factors such that corresponding factors h a v e isomorphic S -a lgebras over ~2, and their related subgroups are o f equal index,.

PRoof. - The subgroups K~= < K~_I and Lt<_--Lj_t satisfy the hypotheses of the Four Subgroup Theorem which immediately yields our statements.

10. - T h e J o r d a n - H S l d e r T h e o r e m f o r T - C o m p o s i t i o n C h a i n s .

DEFINITION 10.1 -- A T - s u b n o r m a l cha in

G - - - - L o > L I > . . . > L , . ~ L

is called a T-composi t ion chain f rom G to the T - s u b n o r m a l subgroup L, a n d its factors are called i ts composition factors, i f L~ is a m a x i m a l (TALc_ l~~1) -



normal subgroup of L~_~, that is i f there does not exist a~y (TOL~_~//1)- normal subgroup of L~_~ properly between L~_t and L~, for all i -~ 1, ..., r. Any T-composition chain from G to the T-normal subgroup ~ of G is called a T-composi t ion series of G.

PI~OPOSITIO~ 10.2. - A T-subnormal chain

G----Lo> L I ~ . . . ~ 5 , . - ~ L

is a T-composition chain i f and only i f L~_~ is (T(~L~_~// L~)-simple, that is i f and only i f its factors TAL~_~//L~ are simple for i = 1, ..., r.

PRooF. - Suppose that H is a T-subgroup of G such that Li_~ =>H >--L~. Since Li is (TNL~_~/ / l ) -normal by assumption, H is (T(SL~_~//1)-normal if and only if H is (TAL~_~//L~)-normal (Theorem 4.9), and our Proposit ion follows from Definition 4.10.

TttE JORDA~Y-H~LDER THEOREM FOR T-CoMPOSITIO~ CI:[AINS 10.3. - I f

G = K o ~ K I > . . . > K,.----L

are two T-composition chains from G to the T-subnormal subgroup L, then

(1) r = s.

(2) There ex, ists a permutation 7: of ( 1, ..., r} such that the S-algebras

~2(TAK~_~//K~) and ~2(TAL~(1)_~//L~,))

are isomorphic and

I g ,_ t :g , [ ---- t L~.,_~ :L~., i

holds for all i = 1, ..., r.

The proof immediately follows from the Refinement Theorem 9.2.

REFERENCES

[1] I. SOEUR, Zur Theorie der ~infach transitiven Permuta~ionsgruppen, Sitz. - Ber. Preuss. Akad. Wiss . , l :)hys.- Math. K1. 1938, pp. 598-623.

[2] O, TAMASCHKE, Ringtheoretische Beha~dlung einfach tra~sitiver Permutationsgruppen. Math. Z. 73, (1960), lap. 393408.

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0 . TAMASCHKE: On the Theory o] Schur-Rings 43

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