Table Of ContentISTITUTO NAZIONALE DI ALTA MATEMATICA
S Y M P O S IA
M A T H E M A T I CA
VOLUME I
ACADEMIC PRESS LONDON AND NEW YORK 1969
ODERISI « GUBBIO
Published by Istituto Nazionale di Alta Matematica Roma
(GJ 1969 by Istituto Nazionale di Alta Matematici
Distributed throughout the world by
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Gubbio, Tipografia Oderisi Editrice - Aprile 1969
Symposia Mathematica
Convegni del Dicembre del 1967 e delPAprile del 1968
I lavori del primo di questa, die si spera essere una lunga
série di convegni indetti dalPIstituto Nazionale di Alta Matematica,
sono stati aperti dal Commissario straordinario professor Giuseppe
Scorza Dragoni con le parole seguenti :
Signori, vi prego di non volermene se non posso lasciarmi
sfuggire Poccasione di un piccolo discorso. Nei riguardi delPIstituto
Nazionale di Alta Matematica credo di avere sempre agito sine ira
et studio. E forse proprio per questa mancanza di rancore e di
indulgenza, un giorno mi vidi improvvisamente posto per bersaglio
alia contraddizione. Da allora imposi alPIstituto un po ? di sordina.
Pub darsi si sia detto e pensato ehe ardire e franchezza non avessi,
ehe viltà nel core allettassi, non sentendomi ne un Enea ne un Paulo.
Ma veramente ebbi altri motivi : e si riassumono in quello ehe ritenni
opportuno agire cosl nelPinteresse delPIstituto (è inutile star qui a
chiarire in ehe senso). Adesso prolungare ulteriormente questo quasi
silenzio potrebbe tradursi più in un danno ehe in un vantaggio ;
epperö provvedo, domandando ai matematici una collaborazione
maggiore e più piena. E la mia chiacchierata è al punto nel quale
la volevo : nel punto di ringraziare coloro ehe mi hanno aiutato e
mi aiuteranno a non far rimpiangere troppo Popera organizzatrice
di Francesco Severi e dei suoi successori, sia ufficiali sia ufficiosi.
Il ringraziamento naturalmente si rivolge anche ai presenti ; ed al-
PIstituto Matematico delP Università di Borna, il quale in questo
momento ci ospita in una delle sue aule. Con ciö la mia parte è
finita. Ora tocca al professor Zappa, il curatore tecnico del convegno.
AN EXTENSION OF OEOUP THEORY (*)
OLAF TAMASOHKE
We are dealing with a mathematical structure which originates
from the theory of #-rings, which is based on the group structure,
and which has been invented to provide group theory with new,
yet elementary, tools for investigating the structure of groups.
brings are certain subrings of the group rings of finite groups.
They were discovered by I. SOHUR [1] and have been used mainly
in the theory of finite permutation groups to prove remarkable
results in an elementary fashion. WIELANDT has introduced the
name of $-ring (i. e. Schur-ring) ; he has developed the ideas of
SOHUR, he has expanded the theory of SOHUR by new ideas and
concepts of his own, and has given important applications of this
theory. We refer to WIELANDT'S book [7, Chapter IV].
Several years of work on brings and on a generalized character
theory on finite groups ([2] and [3]) have recently led the author
to a new view of the theory of Ä-rings. The very simple idea is to
look at #-rings not as a special type of rings but as a mathemati-
cal structure in its own right, that is to produce a notion of $-ring
homomorphism so as to obtain a category. It appeared that this
concept and the theory arising from it can easily be generalized
to arbitrary groups. It is the purpose of this paper to give an
outline of this theory.
1. The Definition of ^-semigroup.
Let G be a group. The set G of all non-empty subsets of G
is a semi-group with respect to the « complex » multiplication
(X,Y)-+XY=[xy\xZX and ytY).
(*) Conferenza tenuta il 15 dicembre 1967.
6 OLAF TAMASCHKE - An Extension of Group Theory
DEFINITION 1.1. A subsemigroup J of G is called an ^-semi-
group on G if it has a unit element and if there exists a set XT ^ G
such that
(1) Q=V Z.
(2) 6 = Z or ό fl Z = 0 for all ό,ΖίΖ.
(3) Z~l = {g-1 \gtZ}tZ for all Zi XT.
(4) X = U Z for all XiJ.
Τηΐ?έ0
(5) T is generated by XI, that is every element of T is the product
of a finite number of elements of XI.
The set XT is uniquely determined by the axioms (l)-(5). The-
refore we call the elements of XT the J-classes of G. The T-class
of G containing g 6 G is denoted by Z .
g
EXAMPLES. 1. The set of all conjugacy classes of G satisfies
(5)-(l) for the subsemigroup of G generated by it ; 1 6 G is its unit
element.
2. Let H be a subgroup of G. The set R : G : R = [RgR \gZG)
of all double cosets of G modulo H satisfies (l)-(5) for the subse-
migroup G/R of G which is generated by R : G : R and whose
unit element is R. We call G/R the double coset semigroup of G
modulo R. G/H is a group if and only if R is a normal subgroup
of G, in which case G/R is identical with the factor group of G
modulo R. We look at G/R as a factor structure of G modulo R
even if R is non-normal in G. In particular every group can be
considered as an ^-semigroup with the set of all its elements as
the set XT.
Let F be a group, Σ be an ^-semigroup on F, and 5 be the
set of all 2* classes of F.
DEFINITION 1.2. A mapping φ of T into Σ is called a homo-
morphism of the S-semigroup T on G into the S-semigroup Σ on F
if it has the following properties.
(1) (XYyp = X* Yv for all X, IE T.
(2) For every ZtTL there exists an di € 5 such that
Z* = 6 and (Z~'r = ύ~ι .
OLAF TAMASCHKK - An Extension of Group Theory 7
(3) X* = U Z* for all ΧεΤ.
It is obvious how to define epimorphisms, monomorphisms, iso-
morphisms, endomorphisms and automorphisms. If the ^-semigroup
T on G is a group then the homomorphisms of the ^-semigroup T
coincide with the group homomorphisms of T.
For every ^-semigroup T on G we introduce the set T of all
those non-empty subsets of G which are unions of T-classes. T is
a semigroup with respect to the complex multiplication, and it is
closed under taking set theoretical unions of arbitrarily many of
its elements. Every homomorphism φ : T —y Σ can be uniquely
extended to a mapping φ J:—yZ such that φ is a homomorphism
with respect to the complex multiplication and to arbitrary set
theorical unions. So it seems more natural to deal with T instead
of the ^-semigroup T itself. The only disadvantage is the following.
Every homomorphic image Τφ of an ^-semigroup T on G is
an S semigroup on the group G^ . Therefore the ^-semigroups form
a category which properly contains the category of all groups,
whereas the category of the structures T does not contain the
category of all groups, a fact that conflicts with our objective to
obtain an extension of group theory. A first step to incorporate
group theory into the theory of ^-semigroups is to link every S-
semigroup T on G with certain subgroups of G.
2. T-subgroups.
DEFINITION 2.1. A subgroup H of G is called a T-subgroup of
G if H is a union of T-classes, i.e. Ht J.
For every subset A of G we denote by < A > the subsemigroup
of G which is generated by A. With every T-subgroup H of G we
associate two ^-semigroups,
an ^-semigroup TH= (ZtTÎ\Z ^H) on H,
an ζδί-semigroup TGIH = (HZH\ZtTi) on G.
Note that Tir^T, but T^iTnO/JÎ.
THEOREM 2.2. Let H and K he T-subgroups of G. Then H (λ Κ
and (H, K) are T-subgroups of G. Therefore the set of all T-subgroups
is a sublattice of the lattice of all subgroups of G.
8 OLAF TAMASCHKE - An Extension of Group Theory
3. T-iiormal Subgroups.
THEOREM 3.1. Let K be a J-subgroup of G. The following sta-
tements are equivalent,
(1) KZ = ZK for all Z 6 U.
(2) XJGIK^JGIK and JGjK X^JGjK for all XtJ.
DEFINITION 3.2. A subgroup K of G is called T-norinal, if K
is a T subgroup of G such that 3.1 (l)-(2) hold.
THEOREM 3.3. Let H be a Jsubgroup of G, and K be a J-nor-
mal subgroup of G. Then
(1) HK = KH is a J-subgroup of G.
(2) HK is a jQ,s-normal subgroup of G.
COROLLARY 3.4. Let H and K be J-normal subgroups of G.
Then UK is a J-normal subgroup of G.
It is an open question whether the intersection of two T-nor-
mal subgroups is itself T-normal, in which case the set of all T-nor-
mal subgroups would form a modular sublattice of the lattice of
of all subgroups. Using representation theory we can prove the
following result.
THEOREM 3.5. If G is a finite group and H is a subgroup of
G, then the set of all G/H-normal subgroups is a modular sublattice
of the lattice of all subgroups of G.
For double coset semigroups we can give some further state-
ments.
THEOREM 3.6. A subgroup K of G is G/H-normal if and only
if K =Η{ΚΓΐ Κη for all g E G.
THEOREM 3.7. If K is a G/H-normal subgroup of G and
H :g L :g G, then the normalizer °fl (L) of L in G is contained in
G
the normalizer °ii (KL) of KL in G.
G
COROLLARY 3.8. If K is a G/H-normal subgroup of G, and if
L is a normal subgroup of G containing H, then KL is normal in G.
OLAF TAMASCHKE - An Extension of Group Theory 9
Another result shows in which way the normal subgroups are
embedded into the ordered set of all ß/JT-normal subgroups of G,
THEOREM 3.9. Assume that H ^Ν ^.K :g G and let N be nor-
mal in G. Then K is G/H-normal if and only if K is normal in G.
4. The Homomorphism Theorem.
Let φ be a homomorphism of the # semigroup T on G into
the ^-semigroup Σ on F. The ^-class o of F containing 1 6 F is
t
the unit element of Σ.
DEFINITION 4.1. Ker«?= U Z is called the kernel of w.
Zfp=6
i
THE HOMOMORPHISM THEOREM 4.2. Let φ be a homomorphism
of the S-semigroup J on G into the S-semigroup Σ on F, and set
K = Ker φ. Then
(1) Zl= Z* if and only if Z K = Z K.
x y
(2) K is a J-normal subgroup of G {and hence the sets ZK=KZ
= KZK, ZtTL, are the Te/x-c^sses of G).
(3) The restriction of φ to JGIK is an isomorphism of the S-semi-
group JGJK on G onto the S-semigroup J v on Grp .
THE CANONIC EPIMORPHISM THEOREM 4.3 Let K be a '[•nor-
mal subgroup of G. Then
(1) φ : X—y XK is an epimorphism of the S-semigroup J on G
κ
onto the S semigroup ~\G\K on G.
(2) Ker φ = K.
κ
We call q> the canonic epimorphism or the projection of T onto
R
JG/K J and we call J JK the factor S-semigroup of T modulo K. In
G
order to keep in accordance with the notation of group theory, we
write JGIK = Τ/Τκ .
Theorems 4.2 and 4.3 show that the T-normal subgroups of
G are exactly the kernels of the homomorphisms of T.
10 OLAF TAMASCHKif i- An Extension of Group Theory
5. The Isomorphism Theorems.
THE FIRST ISOMORPHISM THEOREM 5.1. Let K be a J-normal
subgroup of G, and let L be a J-subgroup of G such that K rg L lg. G.
Then
(1) L is 7 normal if and only if L is J IK-normal.
G
(2) If L is J-normal, then (J/J )/1L = T/Tz, .
K
THE SECOND ISOMORPHISM THEOREM 5.2. Let H be a J subgroup
of G and let K be a J-normal subgroup of G. Then
y
(1) H Π K is a 7H normal subgroup of H.
(2) K is a JHK -normal subgroup of UK.
(3) X—y XK is an isomorphism of the Ssemigroup JH/JHHK on
H onto the 8-semigroup JHK/TK on LIK.
6. 7 subnormal Subgroups.
If R is a T-subgroup of G, and if K is a T - subgronp of H,
R
then we write TSJK instead of (TH)HjK (cf. section 2).
DEFINITION 6.1. A subgroup L of G is called 7-subnormal if
there exists a finite chain
o = i ^ X g . . .è L = L
0 1 r
of J-subgroups of G such that Li is a ~\L_ -normal subgroup of Xi_i
Î 1
for every i'== 1,... , r. Such a chain of subgroups is called a 7-sub-
normal chain from G to L, and the S-semigroups 7i _ /i on L^i
i 1 i
(i = 1,... r) are called its factors ; it is called a 7-composition chain
?
and its factors are called its composition factors if L_ > L and if
i 1 t
there exists no 7L _ ' normal subgroup of -L;_i properly between L^i
i 1
and Li for all i = 1,... , r.
The two Isomorphism Theorems yield, as in group theory,
THE FOUR SUBGROUP THEOREM (ZASSENHAUS7 LEMMA) 6.2. Let
K K,L L be J-subgroups of G. Assume that K is a J normal
0J 01 0 K
