General group extensions

General group extensions

Citation for published version (APA):

Kerkhof, van de, H. P. J. (1968). General group extensions. Technische Hogeschool Eindhoven.

https://doi.org/10.6100/IR59752

DOI:

10.6100/IR59752

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Published: 01/01/1968

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GENERAL GROUP EXTENSIONS

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE HOGESCHOOL TE EINDHOVEN OP GEZAG VAN DE RECTOR MAGNIFICUS DR. K.POSTHUMUS, HOOGLERAAR IN DE AFDELING DER SCHEIKUNDIGE TECHNOLOGIE, VOOR EEN COMMISSIE UIT DE SENAAT TE VERDEDIGEN

OP VRIJDAG ZS JUNI 1968 TE 16 UUR

DOOR

HENDRICUS PETRUS JOHANNES VAN DE KERKHOF

Contents

Index of notatien - - - -1. Introduetion and summary 2. Construction

3. S-functions 4. Neargroups

5.

Equivalent S-extensions - - -

-6.

A system of two-sided representatives - - -7' The the

8.

A. B.

c.

D.

conneetion between the S-functions and

transfer - - - -

-Schreier-extensions - - -

-Pure extensions

S-extensions with a group

Frobenius-extensions - - -Page 8 1 1 1 9 25

35

3T

42

49

55 57 61

62

9. Generalisation of the construction of sectien two 65 10. A construction of an S-extension from a veft

Schreier-extension and a given pure extension - - 70

11. Group products 77

Bibliography - - -

-Samenvatting - - -

{a,b, ••• } {x

I

P} N

z

Q R A><B o (A) \ (a, b) ajbi iff ( 1-1} +:S~T a+ ab.p a (b.p) +ID +ID:D-+K

,..,

K.p

{r,o} G a G/H [ G:H]

Index of notation

empty set

set whose members are a,b ••• set of all x such that P is true set of natural numbers

integers rational numbers real numbers cartesian product cardinal number of

A

set ditterenee

greatest common divisor of a and b a divides b

if and only if one to ene

Ijl is a tunetion from S into T the image of à under the tunetion Ijl

the image of ab under the tunetion Ijl

the product of a and bijl tunetion Ijl restricted to D

if lji:S+T and DeS, KCT then +ID:D+K is lj/ID with range restricted to K

if lji:S..,.T and KCT then K~ = {aiaES,aljiEK'} neargroup consisting of a set

r

and a binary operatien 'o

subgroup of the permutation group G fixing the letter

a

quotientgroup index of H in G

G' <

s>

sr

Core(A) Ker(<)l} (A)

c

0(A) A ut ( G) Inn(G) Lat(H,G) Ch(g) commutator subgroup of G subgroup generated by S

symmetrie group on a set

r

also i f

r

is infinite greatest normal subgroup contained in A

kernel of 4>

normalizer of A in G centralizer of A in G

group of automorphism of G

group of innerautomorphisms of G

latice of all subgroups of G containing H

One

Introduetion and summary

1.1. If Gis group a:ïd a normal ubgr :_;f G) a unique ....

Iy d ined quotien0 oup Q corr ponds to G and K. It may

as if from t groups K gr G may be foèlnd

back. Th pr :em may be formul ed ewhat differ ntly as

fel ows.

Let two groups K and Q be given. Determine all the groups

cent n ng as a normal subgroup, o that G/K is isomorphic

w h Q.

0. Schr ier

I

1 '] wa the irst to give a salution of this

-;:>roblem, o that from now on a group G, which bas b n

construct-ed in the above-mentionconstruct-ed from K and Q, will be called a

Schreier-extensi n of K with

Q.

nerally s e every group ontaining a group A as a

sub-group is called an extensicn of A. R. Baer [ 1] bas amply

st ied these extensions. They will be called general extensions.

Let a group A and a set

r

be given, so that

Anr

=

{e} (e is the

identity of A). The purpose is to construct all the groups which

have A as a subgroup and

r

as a system of representatives of

the right cosets of A in G.

So the final result is to become a group G on the set A U Aa;

aEr\ {e} ea is also indicated by a. Therefore we require bath A and r to be subsets of G.

The construction as described by J. Szép [ 14] is given, without

enter into det s. In it an element of A is indicated by a

small Latin letter and an element of

r

unequal to e by a small

Greek letter. Put AUf T. If i t is possible for every pair as

.

'

Pairs ab with aEA a.nd bEA are multiplied as if they were elements of A and pairs aa are left as formal products. It stands to reason to introduce two functions as:rxT7A and

a6_{~rxT+r and to write as}

₌

0_s.a5_{• The point between as and as} is written~for clarity.

These functions are called S-functions. It then still remains to impose such conditions on these S-functions that G becomes a group. The requirements which are to be made for the S-functions in order that e becomes the identity of G and that every element of G has an inverse, J. Szlp ~alls the initial conditions of the 8-functions. The requirement that tbe eperat-ion is associative, results in

6

functional equations •

Tbe outcome as publisbed in J, SzSp [ 14] is reproduced bere. Initial conditions:

e.

for aer,

a a,

as _{6 ... a = e, for aer, ser,}

Vaer, aser: Ba = e

and 8 is uniquely determined,

( aa) (bil) =(a(ab))B = a((ab)S), for aer, ser, aEA, bEA,

ea a, for aer,

a a = bS

...

a

₌

b and a =

_s.

for aEA, beA, aer, ser.

THEOREM (SZ!:P). A set satisfying the above mentioned

con-ditions is a_ group, iff for the functions

as

and

a5

the

fol-lowing conditions hold:

a a ( ab) a _a. Cl

b,

for

aer, aEA, bEA,

(aa)b

₌

_a ab

•

for

aer, a.EA, bEA,

(as)a

₌

((l

B

a)S , a

_for

_{aer' ser, a.EA,}

(a 8 )Y

₌

(a

B

Y)6 , y

_for

_{o:er, ser, rer,} as

a( Ba.). a Ba.

all. a (sa).

_for

_{aer, ser, aeA,}

a a 6

1\

1.2. In sectien 2 the above-mentioned theerem is proved by choosing ancther construction. See W.Peremans [ 8] .In his construction Szfip starts with the fact that the elements of ~ are of the form aa with aEA and aEf, then defines a product with use of the S-functions and finally derives conditions in order that the group axioms are satisfied, of which the axiom of

associat is the most difficult. We turn the ether way

round and take a starting point where the associativity is guaranteed from the beginning, whereas the fact that the elements of G may be written uniquely in the form aa is archieved by imposing conditions on the S-functions.

The group G is generated by A and

r,

so it is plausible to start with the free group generated by the set T. For teehui-cal reasans it is preferable to choose a free semigroup F instead of a free group.

Again the S-functions as and as are introduced and in F a set of relations is chosen to the effect that a word of the form WabU becomes equivalent to WcU if ab = c in A and a word of the form WasU becomes equivalent to W.as.as.U (Wand U denote arbitrary vords of F).

The quotient sem oup with respect to these relations 1s formed. As a matter of course this semigroup is associative. The requirement that every element of this quotient semig~oup is an equivalence class containing exactly ene word of the ferm aa together with the requirement that the semigroup is a group gives in a natural vay the initial conditions and functional equations found by Sz

1.3.

In sectien 3 the functions and are examined. The base-theerem of this sectien says that the mapping ~:G+Sr defined by a(gn) = (ab)S for aEf, g bS, bEA, 6Ef, is a permutationrepreseritation of G on

r ,

so that the mapping a+as for any sET is a permutation of the set r.

1.4.

The function a 6 :rxr+r is a binary eperation on r with e as identity and having the property that any element has a

unique left inverse. This oper~tio~ however, need not be

associative because (a6 )Y

=

(a Y)6 • A structure consisting of the set

r~and

an operatien with the properties mentioned above

is called a neargroup. In the case of a Schr ion the

neargroup is the quotient group G/A.

Similarly as in the case of the Schreier-extensions, the general extension problem can now be considered to be the construction of all the groups G with a given subgroup A and a given near-group

r.

In sectien

4

this is worked out further.

1.5. The conception "equivalent S-extensions11

, which logic-ally results from ~he introduetion of another system of repres-entatives is discuseed in section

5.

First of all it is noticed th.at, in contrast with Schreier-extensions in which equivalent extensions have the same quotient group, equivalent s-extenmons ca.n have non-isomorphic neargroups.

Passing on to a suitable equivalent extension the S-functions can sometimes be greatly simplified.

A necessary and sufficient condition for an S-extension to be equivalent to an extension for which

VaÉr,vser :as=

e i s deriyed, Analogous to the case of Schreier-extensions these 6-extensions will be called splitting. It appears that in this case the neargroup

r

is a group and G = Ar with Anr = {e}.

From this it fellows that the extensions studied by G.Zappa [ 16] form a special case of the S-extensions discuseed bere.

1.6.

If A is a subgroup of G and r a system of representat~v­ es of tbe right cosets (s.r.rJ of A in G,

r

is - in ge6eral -no system of representatives of the left cosets (s.l.r.) of A in G. If, however,

r

is a common system of representatives of tbe right and left cosets,

r

is called a s.t.r.

Because as is derived in

2J6.,

with every

aer

an

äer

is associated, so tha.t äa = e, a mapping 4> of

r

into

r

may be defined by a+

=

ä.

ecti n 6.

In 6.1.' r l S shown to an s . t . r . i f f '' l S (1-1) and onto or

formulated differently i f f every ele5ent of the neargroup r bas

right in7erse.

Moreover, it is pr ed,

mutation of A for all ae

that q, is (1-1) and onto.

the condition that, a is a

per-is equivalent with t ondition

1s immediately cl that both with Schr ier and with

Zappa-extensions this c ndit on is fulfilled because r, these

exten-ns the neargroup is a group. In the case o

Schreier-ens ons 0 l an automorphism of A.

Furthermore i t is inve igated wether for a given group G and

a given subgroup here exists an s.~.

6.8.

that for the existence f an s.t.:r. It will be proved i

i t is necessary sufficient that th

izer of An is qual t the order of

ord

a-of the a-stabil-abilizer of Arr, or,expressed differently, that in any

are as many right as left cosets of A.

ouble coset AaA there

The theorem proved H.Zassenhaus [ 17] that A has an s . t . r .

if the index or 6. 8. l·loreover

in G is finite, fellows from the result of

a most trivial consequence of 6.8. is that

any finite subgroup of a group has an .r.

In

6.14.

an example of a subgroup without an s . t . r . i s given.

According to 6.16. and 6.17. "<Pis (1-1)" and ".pis onto" are

both sufficient conditions for the existence of an s . t . r . In 6.18. and 6.19. i t is shown that, "Vaer:ao is (1-1)" and "VaEf:ao is onto" are bath of them also sufficient in order that A bas an s . t . r .

1.7. Insection 7 the conneetion between the transfer of G in A and the 8-functions is discussed.

If G AB with AnB ~ {e} in whicb A is a subgroup and B a

normal subgroup of G, B is called a normal complement of A. Some theorems will be proved for S-extensions of a finite

an Abelian Hall subgroup of G). (8o A is

7.5.

IfVaEA:a-1

ra = r, A has a normàl complement.

If A is a p-group and NG(A) = CG(A), A has a normal complement (Burnside).

7.11. If A is a cyclic p-group and NG(A) =A, G1 _{is a normal} complement of A.

7.11. If o(A) is a prime and NG(A) A, G is solvable.

1.8. While in sectien 7 attention is paid to finite 8-extensions, in sectien 8 some special 8-extensions are consid-ered in which other restrictions then finiteness are supposed. 8A contains a short discussion of those 8-extensions for which . Core(A) is as great as possible, i.e. Core(A) = A. These are

clearly the 8chreier-extensions. In order to illustrate the capneetion between the S-functions and the factorsets and

aut~morphisms accuring in a Schreier-extension, some theorems on Schreier-exten~ions will be proved.

In 8B the other extreme case viz Core(A) = {e} will be treat-ed. These extensions are called pure-extensions.

Seeing that in this case w is an monomorphism we conclude that a pure-extension is isomorphic with a transitive subgroup of Sr. Therefore it is possible to describe all the pure-extensions belonging to a given neargroup· r.

Another consequence of the above is, that the 8-functions have to fulfil fewer conditions. Also S-extensions for which the neargroup r is a group wil.l be discussed ( 8C), Although in these group-extensions the group r is not necessarily a sub-group of G, still it is possible to show that some theorems, about the solvability of G, holding for factorizable groups, also bold tor these group-extensions.

In 8D it is not only supposed that Core(A) = {e} but the

much stronger condition is made that Vaer\ {e}: (aa = a • a = e), so that for a ~ e the a-stabilizer of Aw consists of one

element. Limitation to finite groups results in G to be a Frobenius group with a Frobenius-kernel, which is a normal complement of A.

The above assertation will only be proved in the case of A being olvable. R.Shaw [ 13} proves the same theerem in another way. The theerem holds also good if A is a non- olvable group, of this fact no character-free proef is known. For a proef ,.,jtb 0he use of group charac"ters see B. Huppert l 6].

1.9. A generalisation ·Of the construction of sectien 2 will be discussed in sectien 9. In this generalisation lv

...

lS sup-posed that A lS given by a generating set T

1 and nuwber of defining relations. G is an S-extension of the group A with the set

r.

The purp se is t construct the group ~ with subgroup A from a gener ing set T

1U instead of AU •

This construction wi 1 be use in sect on 10 1 1 •

1.10. Now a general S-extension will b reduced t a

Schreier extension and a pure-extension, which elucidates the fundamental character of these two extension

p G is an S-extension of A with

r,

A is a S hr ier-extension of Core(A) vith An.

Core(A), An and the S-function belonging to the construction of A from Core(A) and An are supposed to be en.

Ge is a pure-extension of An vith rn. Also the S-functions belonging to the construction of Gn from An and fn are sup-posed to be given. As a generating set of the semigroup, of which G is a quotient group, T Core(A)UAu will be taken. It will be shown that in addition to the functions given befare only one function: aut:ruxT~Core(A) is necessary to construct G. The fundamental equations for this function will be derived insection 10.

1.11. a group A, a group Band a cammen subgroup C of A and B are supposed to be given. The aim of the construction described in sectien 11 is to construct all the groups G, sa that G

=

A'B' where A' is isomorphic with A, B' isomorphic with B and A'nB' isomorphic with C.

The metbod used in this thesis is based on section 9.

Also by L. Redei and J. Sz~p [ 10] a construction is described. G. Casadio [2) describes a construction for the case that C is. a normal subgroup of G.

Two

Construction

2.1. DEFINITION. Lets be a set. Let W be thesetof all non-empty finite strings ("words") of elementsof S ("letters"). The free semigroup F generated by

s

is the structure with W as set and juxtaposition as operation.

2.2. REHARKS.

1. The associativity of the eperation of F i s trivial. 2. The elements of S are identified with the corresponding

ene-letter words of F.

In F a set I of relations X Y, with XeF and YEF, may be chosen. In fact I is a subset of FxF.

2.3. CONSTRUCTION OF A CONGRUENCE RELATIONS RI WITH RESPECT TO I .

{(U,V) JueF,VEF,inENU{ol and ViENU{o} with < 3:U.EF

~ with U 0 =U, 3:W.EFU{~},iZ ~ or (Y.,X.)ei, l l

Un

=

V such that vieNu{o} with o< <

{ l1i}

,ax

and iY such that (X.,Y. l l W.X.Z. and U. W.Y.Z.}

l l l l l ~ ~

I

2.4. REMARK. It is clear that ICR

1 and that RI is an equi-valence relation. Moreover if U

1eF, U2EF, V1eF, V2eF,

(U ,V )ERI, (U ,V )ERI' then (U U ,V V )ERI' i.e. RI is a con-_{t l 2 2 . 1 .t: 1 2} gruence relation. So in the quotient set of F with respect to RI multiplication may be defined by multiplication of represen-tatives of the equivalence classes with respect to RI. The quotient set with this multiplication is the quotient semigroup

2.5. DEFINITION. The quotient semigroup G of F with respect

to (the congruence relation) RI is càlled the quotient

semi-group of F with respect to (the set of relations)

I .

2.6.

REMARKS.

1. Different elements of S may happen to be identified inG. So we cannot say that G is. generated by

s.

2. This construction must nat be confused with the case that the elements of S are considered as indeterminates, where in nrelations" substitution is allowed •. In our case, if ab 5 ba is written, this permutability is only required for these elements, for which the relatipn is given.

3. ~very semigroup with generating set S is isomorphic vith

a quotient semigroup of

F.

Let A be a semigroup vith identity e. Let ab denote a two-letter word but (ab) the ene-two-letter word consisting of the letter, which is the product in

A

of the elements a and b of

A.

Let

r

1 be a set and f1(')A = r/J, Put r

=

f1U{e} and T = f1UA, Let the following mappings be given:

as:r

1 xT+A vith aer 1 ,sET and aseA, a8:r xT+r

.) vith aer1 ,sET and a

8_er.

Let F be the :tree semigroup with T as generating set. The following set of relations I in F is taken:

V~eA,VbEA:ab 5 (ab), Voer ,WsET:as 5 as.a.8,

1.

Vae

r

₁ :

ea

5 a,

Let G be tbe quotient semigroup of F with respect to I. We are going to determine necessary and sufficient conditions in order that every element of G ~ontains exactly one word of the form ap with aeA and per.

For that purpose a standard reduction will be defined. We first extend the functions as and as as follovs Ps:fxT+À with per,seT and PseA,

p8:rxT+r with per,sET and pser, by putting

2.7. DEFINITION of a reduction-step.

Let r2 = tap

I

aEA, pEr}.

A reduction-step f:F\Q~F is defined by:

aWf eaW for aEr ,WEFU{<j')},

af = ae for aeA,

abWf = (ab) W for aEA, bEA\ ~e} ,WEF;J{<j')},

apsWf = (a. s)p5 .W for aEA, f,sET,WEl''U{<j')} •

2.8. REHARKS.

1. Taking p

=

e in the ast line of • 7· we get

aebWf (ab)eW fr:·r aEA, bEA, WEFU{<j')} ,

aepWf apW for aEA, DE WEFU{ 'IÎ}

2. If gEG, UEg and U~Q, then UfEg.

3. Let r2

₌

ff' and n fn-lf, then VWEF\rl, :irn with WfnEQ.

So every gEG contains at least one ement of

2.9. DEFINITION OF STANDARD REDUCTION.

A stanaard reduction S:F-Q is defined as fellows: For We : WS

w,

for WE:F'\ : WS = Wf'n •,vi th the n for which \Hne:.l.

2.10. REMARKS.

1. If gEG and UEg then USEg.

2. WS

=

WfS for WeF\rl.

2.11. THEOREM.

1. UVS USVS for UeF and VEFU{$}.

2. cUS c(US)S for cEA and UEF.

PROOF. From the definition of S it follows that i t suffices

to prove:

1. uvs UfVS for UEF\Q and VUFU{~}

2. cUS c(Uf)S for cEA and UEF\Q.

The only case in which the first of these two is not trivial

is U a with aEA. We then apply mathematica! inducation with

US= UfS (2.10.a). If V~~ we put V

or V

=

bV

1 with bEA\ {e}, V

1

EFU{~}.

apV

1S aepV1S

=

afpV1S, abV

1S

=

(ab)V1S = (by induction)(ab)fV1S

=

(ab)eV1S

=

aebV1S afbV

s.

1

Before proving the second equality we first remark that abWS

=

(ab)WS for aeA, bEA, WEFu{~}, for b ' e this fellows trom 2.10.2 and for b

=

e from 2.11.1. Now 2.11.2. follows from the associativity in A and 2.10.2.:

caWS = ceaWS

=

c(aWf)S, caS

=

(ca)S

=

(ca)eS

=

caeS cabWS (ca)bWS

=

((ca)b)WS

c(a.bWf)S,

c(af)S,

(c(ab))WS = c(ab)WS

capsWS = (ca)psWS = ((ca).Ps)p8WS

=

(c(a.Ps))p8WS

=

c(a.Ps)p8WS

=

c(apsWf)S.

2.12. THEOREM. Every gEG contains exactly one element of ~ iff \faEf₁, \f(X,Y)EI : aXS = aYS.

PROOF. Every gEG contains at least one element of~ (2~8.) Suppose every gEG contains only one element of 0 then

aXS

=

aYS because (X,Y)EI • (aX,aY)ERI • (aXS,aYS)ER

1•

Nov suppose \faer

1 , Y(X,Y)ei:aXS

=

aYS.

Y(X,Y)EI:XS

=

YS.

Y(X,Y)EI:eXS

=

.e(XS)S

=

e(YB)S

=

eYS.

Y(X,Y)EI, YaEA:aeXS

=

a(eXS)S

=

a(eYS)S

=

aeYS.

y(X.Y)EI, YaEA, YaEf :aaXS = a(aXS)S = a(aYB)S = aaYS.

1 .

y(X,Y)EI, YWeF:WXS = WSXS = WSYS

=

WYS. y{X,Y)EI, YWEFU{~}:WXS

=

WYS.

\f(X,Y")EI, yWEFU{~}, YW

₁

EFU{~}:WXW

₁

S = WXSW₁S = WYSl</

18

=

WYW1S. Y(U,V)ER

1:US

=

VS.

Y(U,V)ER~:(UEO, VEO • US= VS, US= U, VS =V • U= V).

2.13. THEOREM. Every gEG contains exactly one element of g iff YaeA, YbeA, YaET ₁ , Y6Ef₁ , V sET:

PROOF. By straightforward calculation from 2.12.

Suppose that every gEG contains exactly one element of and let o:T->G be defined by: sEsa, saEG for sET. Then a is (1-1).

oiA is a monomorphism, because ((ab),ab _I.

Identifying T and its image in G we may consider A as a sub-semigreup of

G

and r as a subset of

G.

From now on we suppose this identification to be made. G now is generated by T and the elements of G are products of elements of T. This product

is written as juxtaposition, the use of brackets for the pro-duct in A no langer is necessary.

Every element of G may be written in the farm aa with aEA, aE~ Mult

aabS

ication in G is given by:

b

a.ab.a B(ab)S with aEA, aEr, bEA, SEr. In G e is a left identity because eaa

=

aa.

It is easy to verify that the equalities in 2.1~ arealso valid if aEr and 13Ef (instead of f

1 ) .

2.14. THEOREM. If and as satisfy the conditions of 2.13. then G is a group iff A is a group and VaEr, ,a~er, such that

e.

PROOF. Suppose G is a group, aEA and bS with bEA and BEf is the inverse of a in G. Then abB

=

e, so ab

=

e and S

=

e, so every aEA has a right inverse in A. Therefore A is a group. If ctEf

1 and bB with bEA and !3Ef is the inverse of a in ~ then

bl3a = e, b. 13 a.Sa

=

e, Sa= e.

Conversely, su_g_pose, A is a group and VaEf

1 , aEf1, such that

~a

=

e. Then (aa)-1 aa e so every element of f

1 has a left inverse. Because also every element of A has a left inverse, it follows that every element of G has a left inverse. So G is a group.

2.15.

COROLLARY.

I f G

is a group then

a e.

=

ae

₌

a

for

aE f! ,

So

a e

=

e

and

a e

=

a •

But

=

e

and

a e

=

a

imply the

last line of

2.13.

2.16. MAINTHEOREM. If A is a group with identity

e,

r

a set

with

Anr

{e},

T

=

Aur, aer, ser, as:rxT~A

and

as:rxT~r

are

functions, then there exists a group G with A as subgroup,

r

as system of representatives (s.r.r.) of the right cosets of

A in G and

as

=

a a a b a(ab), a. = aB a(Ss).a aB •

s

a _e = e, ea _a, e a = e, a _s.a s Bs

ss.

iff

VaEA,VbEA, Vaer,

vser

and

VsET (aa)b ₌ a(ab)

.

'

(aS)s S 's = (a s) S e a = a, a e = e, ea

₌

_a, Vaer, ~~er

with

äa

=

e.

A

group G satisfying the conditions above is determined by

A,r,as

and

as

up to isomorphism.

G is isomorphic with the quotient semigroup of the free

semi-group generated by T with respect to the relations

ab - (ab) aEA, bEA,

a. s _{aE r\ { e}, sET,}

as - s.a

Three

S-functions

n this sectien a definition of permutati scrnorphisrr, and

of similar permutationrepresentati i given.

3 . . contains a criterium o determine he similarity f two

permutationrepresentations. In .1. a

permutationrepresenta-tion n, playing a fundamental r e in the remaining part of his dissertation, is introduced. An examinatien into the orbits and stabilizers of is made.

If G is an S-extension of A with r it proved in 3.26. that

< f)nA ( {aB!aEf, BEf}>. (< r> is the subgroup generated by

r).

Let SM be the set of all permutations of the set M.

A permutationrepresentation of a group G 1s a homomorphism t

of G onto a group h of permutations of some set M.

Such a representation will be called a representation of G on M. A representati n on M is called transitive (primitive) iff h has this property. A representation is called faithful iff the mapp from G on H is an isomorphism.

3.1. DEFINITION. If Pis a per~utation group on M anä Q a permutation group on N, then ~:P~Q is a permutation isomor-phism from P onto Q iff:

1. t is an isomorphism from P onto Q,

2. a~:M~N which is (1-1), onto and such that ~aEM, ~pEP:a$(p~) = ap~.

3.2. REMARK. As a matter of fact ~ is determined by ~.

3. 3. DEFIIHTION. If ' and TT are permutationrepresentations of the group G on M resp N then ' is similar to TT iff;

Vg EG, Vg EG:g c = =<> g u

=

g u and

I 2 I l 2

•:Gt+Gu defined by gtt gn for gEG is a permutation isomor-phism.

3.4.

REMARK. Similarity is an equivalence relation in the class of permutationrepresentations of a group G.

3.5. THEOREM. If t is a permutationrepresentation of G on M and ~:M+N is (1-1) and onto, then n:G+SN defined by

a~(gu) = a(gt)~ for aEM and gEG is a permutationrepresentation of G on N, which is similar to t.

PROOF. gn:N+N defined by a~(gn) of N.

n:G+SN is a homomorphism, because

a~gt)• is a permutation

a•{g g n) = a(g g t)• = a(g t)(g t)• = a(g t).(g n)

1 2 1 2 l 2 1 2

= a•(g n)(g n), 1 2

So n is a permutationrepresentation of G on N,

~:Gt+Gn defined by gt~ = grr is an isomorphism as fellows immediately from gt~ = •• , (gt)• for gEG.

Then Ijl is à permutation isomorphism from GT onto Grr because a•(gt!jl} = a(gt)• for aEM and geG.

3.6.

REMARK. If n and t are similar permutationrepresentat-ions of the group G on M and N, it fellows immediately rrom definition 3.3. that there exists a (1-l).:M+N of M onto N such that:

v

aEM;v gEG:a.(gt)

=

a(gn) ••

It is well known that:

If A is a •ubgroup of the group G and M the set of the right cosets of A, then t:G+SM defined by m(gt) = mg for mEM and gEG is a permutationrepresentation of G on M.

t is called the natural representation of G on the right co-sets of A in G.

t is transitive and Ker(t)

=

Core(A)

=

the greatest normal subgroup of G which is contained in A.

T i primitive iff A is a max~mal subgroup of

ful iff Core( )

=

(e}.

and ' is

faith-3. 7. THEOREr-1. If G is <:m S-extension of A with , then

r àefineà by ) = ( )6 for aEr, g bB~ bEA, BEf

is a permutationrepresentation of G on 1 similar to the

natu-ral representation r of on the right cosets of A in G.

PROOF. M is he set the right coset of A i G.

$:M+f defined by Aa<P = uE :' is ( 1 - 1 ) onto .. ( ) ( 1 1 D) i's

Aa4> gn, a gn 1

= ,

u A(ct0)8cD

B~

Acx( ge) <P.

Now the theerem fellows from

1' is c the natural represent 1on of the S-extension G on

r.

3.8. THEOREM. If is an S-extension of A with r,~ the

natura! representation of 'J on 1' and A the

norma-lizer of A in , then O:G•Sc (in which Cis thesetof conju-gate subgroups of A in G), élefined by

g-1 ct-1 Aug for G and uEr is a pernutationrepre-sentation of G on

c

similar to n.

PROOF. $:f+C defined by a$ a-1 Aa for aEf is ( 1-1) and

onto because A NG(A).

a,P{ge)

=

a-1_Aa(ge)

=

_g-1_a-1_Aag

₌

_((ab)B)-1_{A(ab)B = a(gn)<jl.}

Now apply theerem 3.5.

3.8a. THEOREH. ·rf G is an S-extension of A with r anà o a homomorphisrn of G with Ker(o)CCore(A) then ro is a systern of representatives of the right cosets of Ao in Go and the image Gn of the natural representation n of G on

r

is perrnutation isornorphic with the image Go11

1 of the natural representation n of Go on r •

1

PROOF. Aofo Go and a

#

B w Acraa

#

AoBo for aer and BEf

So ro is a s.r.r. of Ao in Go. oir= {1-1).

If aer and seruA then: {as)o{{a6)o)

=

{as.a6)o =aso =

0

are the S-functions of Go with respect to the group Ao and the s.r.r. ro,

So (as)o

=

aoso and (as)o

=

ao50• A repeated application results in ((ab)6)o = (aob0)60 for aer,

r,

bEA.

So i f g

=

bS then:

ao(go11 )

=

ao(bo)(So)11₁ : (aab0)60

=

((ab) 6 )o = a(g'IT)o.

I

Therefore g₁'1T = g~rr • g₁0'IT₁ = g₁0'lf₁•

So •:G11+Sr

0 defined by gwf = gow1 for gEG is a mapping. gt1T(gl11_)f

=

gtg~11•

₌

_{gtgla11 t = gt} 0_{1Tt(gzcr1Tt) = gt} 11_f(gl1!1j!).

Let

s••

= e (identity of Sr } then go'IT = e • gaeCo~e{Aa) in Go

a 1

• gECore{Aa)Ó = Core(A) because Ker(o}cCore(A). So g••

=

e • geCore(A) • g'IT = identity of Sr' Therefore w is a monomorphism.

It has been shown already that oir is {1-1) and ao(gww) = a(g'!T)o.

Therefore w: G11+Go11 is a permutation isomorphism. I

3.8b. COROLLARY. The natural representation of G on

r

is

permutation isomorphic with the natural representation of

G11

on

rw

as follows from 3.8a. if we take

o = 11.

3.9. THEOREM. If

11

is the natural representation of G on

the set M of right cosets of the subgroup H and

T

the natura!

representation of G on the set N of the right cosets of the

subgroup K, then

11

is similar to

T

iff H and K are conjugate

subgroups of G.

PROOF. ~et K

=

1~1_Hl

with lEG and r be a system of repre-sentatives of the right cosets of

H

in

G.

It follovs that

•=M~N defined by Ha• =l-1_{Ha = Kl-}1_all-1 _{for aer is}

_(1-1)

_and onto.

Ha.(gT) = l-1Hag = Hag•

=

(Ha)(g1T)• for aer and geG. Let 11 be similar to T and aer. The Ha-stabilizer of G'IT is a-1Ha because {gjHa(g'lT) =Ha}= {gjHag =Ha}= {glgea-1Ha} =

=a-

1

By the similarity of TI and T there exists a (1-1) mapping ~

of M onto N such that Ha (g1) = Ha gn )<P.

If the representative of

K.-•

lS called Y so it fellows y-1_{HY =Kor H is conjugate toK.}

3.10. COROLLARY. Let G be an S-extension of A with

r

and TI

the natural representation of G on • B is a subgroup of u

and I a system of representatives of the right cosets of B with e as representative of B.

T is the natura! representation of G on E. Then n is similar to 1 iff B is conjugate to A.

Let G be an S-extension of A vith

r

and n the natural

representation of G on

r.

The folioving symbols are introduced: For HCG and ACf put:

t;H {o:(h7') !aE6 and hEH}.

I f Ct

=

{ } a , a H . lS wrltten and lf , . . H

=

{ } h , is vritten,

By this notatien a8 for sEAUf obtains a nev meaning, but this is allowed becauie a8 in the nev meaning is the set vhich con-tains a5 in the old meaning as a unique element.

aH is the a-orbit of Hrr so aG

=

r.

3.11. THEOREM. If G is an s-extension of A with r and HCG then AaH

=

AaH for aEr.

PROOF. Because AaHcAaH is evident ve have to prove AaHCAaH, If aEA, aEf, bEA, BEf and bBEH then:

b b

a(ab)!l a(a B)-1 _.(ab)-1 _{.ab.a (3.(ab)BEAab(3CAaH.}

3.12.COROLLARY. If

V

is the set of rightcosets of

A

con-tained in AaH .then o(V)

=

o(aH).

For H is a subgroup of

H

" =

{hjhEH and a(hrr) = a

H 1<

_{" =}

_{{hrr hEH and a(hrr)}

I

=

G and t;Cf put: for yaE6}. a for Vo.EA}.

So Hnà = (Hà)n. (The brackets are used for clearity).

{ } H Hn . . t

If à = a , a and a 1s wr1t en. Gn

e = An because e(gn) = e * gEA for gEG. Gn~ = (an)-1

An(an)

=

a-1

Aan because Gna is the a-stabilizer of Gn.

Hna = a-1

Aani"\Hn.

According to Hall, corollary 5.2.1., [ 5] o(aH) Ha = (Hna)fthH.

Hn [ Hn: a] •

3.13. THEOREM. If G is an s-extension of a finite group A

with

r

and

H

is a finite subgroup of

G

then

o(aH) = [ H:a-1

AanH] for aEr. PROOF. H o(AaH) o(a ) = o(A) o(a-1 AaH) o(A)

o(a-1Aa)o(H) o(H)

o( (a-1

Aa)nH)o(A) o ( (a-t A a )nH)

3.14. THEOREM. If

G

is an s-extension of

A

with

r,

H

is a

subgroup of

G

and

Core(A)cH.

then for

aEr,

1. Ha= (Hna)t,

H

-2. a = a-1_Aai"\H, 3. o(aH) = [ H:Ha].

PROOF.

1. Core(A)CHa because kECore(A)

~

kEH and a(kn) =a.

2. (Ha)n = Hna = (a-1_Aa)wi"\Hn

~ (a-

1_{Aai"\H)n because Core(A)CH} and Core(A)Ca-1_Aa.

So Ha

=~

1

Aai"\H

because

Core(A)ca-

1

~ai"\H.

3. o(aH) = [Hn:(Hahl = [H:Ha] pecause Core(A)cHa.

3.15. COROLLARY.

Taking H

=

A we get:

A (An )+

1. a= an,

2. Aa = a-1_AanA,

3.16. THEOREr-1. If G is an S-extension of A with r, and for heG, ho:G+G is the innerautomorphism af G defined by g(ho)

=

h-1

gh for gEG then:

(Aa)(ao) A(aa) for aEA and aer,

(Aa)(ao)

=

for aEf.

PROOF.

(Aa)(ao) ( )(ua)

a-'a-1_Aaana-1_Aa

=

_(aa)-1_AaanA

=

_A(aa).

a-1 _{a -}1_Aaana-1_{Aa Ana-}1_Aa

=

_Aa.

3.17. COROLLARY. 1 • ATI a ( A a ) a_, •

2 • ~ = A a ..,. ( A a ) f or aE r •

This condition is eviàently fulfilleà if Aa

=

Core(A). 3. If G is an S-extension of A with r and A a = Core(A) for

aE r\ { e} then vae r : 0 ( a ) A o (<XA) for:

o(aA)

₌

[ A: A a]

=[A:~)

₌

o(a ) _A and o( )

₌

o(ëA)

₌

1 •

all

These extensions are called Frobenius extensions of A with r.

3.18. THEOR~1. If G is an s-extension of A with

r

and [A:Core(A)l is finite then 'ifaEr:o(aA)

=

o(aA).

PROOF. o(A11)

=

(A:Core(A)] is finite.

From {~)n

=

(a(Aa)a-1 _)rr

=

_{(a11)({Aah)(a-}1_{11)= (n)((Aa)11)(a1T)-}1 it follows that

o((~)11)

= o((Aa)11).

S o : o ( a A)

=

I

A 11 : ( A a ) 1T] [ A rr : ( ~) rr] = o ('UA) •

3.19. COROLLARY.

1. If Gis an S-extension of A with a finite

r,

then vaer:o(aA)

=

o(aA) for:

f is finite ~ Sr is finite ~ G11 is finite ~ A1r is finite ~ [A:Core{A)] is finite.

2. If G is an S-extension of a finite group A with

r,

then 'ifaer:o(aA) = o(aA) for:.

3.20. THEOREM. If Gis an S-extension of A with r, i t fellows:

A

a SA for aEr and SEr. ~A for aEr.

a

3. aa a a so aACaA for aEf and aEA.

PROOF.

1. evident because aA and SA are orbits of

A~.

a -a a

2. a = e a = ( ~'lf) a = (~ a)a = a a

a a

3. e = (aa)a = <a a)a

.

4.

yEf and YEaA

,.

yEa Ca = A -A

,.

=A y = a -A

,.

y A = a -A

,.

yEa -A

-A _{yEr and} - -A =A A

yEa

..

yEa Ca a •

3.21. THEOREM. If Gis an s-extension of A with rand aEr

1 thenl aA = aA iff :!IgEAa with g2 EA. PROOF.

UA= aA * aEaA 0 :!IaEA:(aa)a

3.22. THEOREM. If G is an s-extension of A with a finite r then :!IaEf\{e} with aA = aA iff 2j[G:Core(A)].

PROOF. r is finite

* [ G:Core(A)] is finite because

[G:Core(A)] = o(G~).

2jo(G~).

*

:!IgEG with (g~)2 is. the identical permutation and g~Core(A).

Suppose {g~)2 = g2 ~ = e~ with g = aa, aEf\{e} and aEA. Then (aa)2_{ECore(A}cA and a :F e. So aA = aA and a :F e.}

Suppose now a2 _~

= e~ with a~ Core(A) then "ll'aEr:(aaa-1 ₎2 _{~ = e~.}

Take aEr such that aa :F a, which is possible because a~Core(A) then aaa-1_~A.

So aaa-1

EAS with SEr\ {e} and (aaa-1

)2ECore(A)cA.

From this it fellows that SA = SA and 8 :F e.

Conversely let :!IaEr\ {e}with aA = aA. Then :!IgEG with g2_EA

and gEAa. So. :!IgEG\A with e(g2 _~)

and e(gTI)

=

aEr\ {e}.

Fron this it fellows that the order of the element gTI of Gn is even, therefore o(Gn) [ G:Core(A)] is even.

3.23. THEOREM. If G is an S-extension of A with r and aEr then

~EN

0

(A)iff

aA {a} and aA = {a}.

PROOF. aEN 0(A)

=

aAa-1

=

_A o:A = Aa (A) ~ a-1_EN 0(A) = (A) Conversely

={a}~ aACAa = ACu-1

Aa. A a -A 0: {a} •

={u}~ aACAa ~ )a-1_AcA(ao:)a-1 _{~ a-}1_ACAa-1 _{~ a-}1_AacA.

3.24. REMARK. From the above proef it fellows that:

= {a} ... Aca - ) Aa.

3.24. THEOREM. If G is an S-extension of A with r, aEr, and [A:Core(A)I is finite then aA ={a} iff A= a-1

Aa iff aEIT

0(A).

PROOF. aA

=

{a} • aAcAa • aTI(An)cAn(an) • an(An)(an)-1

CAn • ~ o:n(An)(an)-1

₌

_{An (because An is finite) • o:Aa-}1_{n An •} => n-1_{Aa = A(because Core(A)cAncxAa-}1

) • aEr1

0(A) • aA ={a} (3.23.) 3.25. THEOREM. If G is an S-extension of a finite Abelian group A with a finite set

r,

then aA = {a} • VaEA:aa = a iff

N

0

(A)

=

c

0

(A).

PROOF. Let A be Abelian, bSEN

0(A) for bEA, SEr. It fellows that bSA(bS)-1

A

a a and

and Va.EA: 13 a =a"" VaEA:Sa aS • bSa(bB)-1 _bSal3-1

b-1

a • bi3EC

0(A) • N0(A)CC0(A) = NG(A) = (A) because C

0(A)cN0(A) is trivial.

Let conversely A be Abelian and finite, N

0(A)

=

c

0(A) and aA = {a}. It fellows that aAa-1_{cA • o:Aa-}1 _{A because A is} finite. So:

A

a = a "" VaEA: a = Cl

a.

G is a group and D is a subset of the set G. The subgroup of

G generated by D is written ( D) •

3.26. THEOREM. If G is an s-extension of A with

r,

it

fel-lows:

1. <

r>

{aa!ae< r>nA, aEr}, 2. <

r> nA

< {a.

e

I

aE

r , ae r }

> •

PROOF. 1. Obvious.

2.

Put<

r>nA

=Hand< {0

Biaer, ser}>

=

K.

KCH because 06 = aS(aS)-1

• What fellows shows that also HeK.

6 By

aS • .a y 0_(!3y).0

(Sy),.. a(By)EK for all aEf, SEf and

yer

b-1

ab-1 _{=(a b)-}1 _{=<- a((Sy)-}1 _{)EK for all aEf, SEf and yEf,}

Suppose gEH then g is a product of elements of

rur-

1 •

From aS = 0

S.a 13 for aer,

ser,

- (13 )-1

aS-1 _{= a((Bs)-}1 _{).a. S .]"}

for

aer

and

eer,

i t follows that g

=

aa with aEK and aEr. Because gEHCA, a

=

e, so gEK.

3.27. DEFINITION.

A

group G is factorizable iff G

=

AB in

which A and B are subgroups of G and A and B proper subsets of

the set G.

3.28. THEOREM. If G is an s-extension of A with

r :f

{e} and

<{aB!aer, !3erl> :/:A then Gis factorizable in G

=

A<r>.

PROOF.

r

:f:

{e} • A:/: G and <

{aBiaer, serJ>

". <

r>

:f:

G •

Four

Neargroups

In order to come to an agreement between the S-extensions and the classical Schreier-extensions in which A is a normal subgr up of G, the neargroup-idea is introduced.

A neargroup is a non-associative structure, indeed possessing an identity and in which each element has a left-inverse. In the case of an S-extension the neargroup takes over the part played by the quotientgroup in the Schreier-extension theory.

4.1. DEFINITION. A neargroup is an ordered pair {M,®} such that lvl is a set and 0 is a binary oparation on M so that: a. 3: eEH such that lfaEM: e0a aee a,

b. V aEM, 1f BEM, 3: yEM: y ® 8 a ,

c. VaEM, YBEM, YYEM:aey

B.

The element e is called the identity of {M,e} and the uni-que element a such that aeB

=

e is written as

B·

A neargroup is a group iff the neargroup is associative.

4.2. DEFINITION. A homomorphism of a neargroup {M

1 ,e} into

a neargroup

{1\ ,

a} is a <1>: M

1 +M2 such that

YaEM

1 , YBEM1 : (aeB)<j>

=

a<j>aScp.

If 4> is (1-1) the homomorphism is called monomorphism. If <1> is onto the homomorphism is called epimorphism.

4.3. THEOREM. If 4> is a homomorphism of 0\,e} into {M

2 ,ll},

e, the identity of {M

1 ,®} and e2 the identity of {M2 ,a}, then

PROOF. e

2a(e1 4>) = e1 4>

a<j>a(a<j>)= (äea)<j>

(el eel )<j> el .pa(el 4>) ~ el 4>

el 4> el ~ a<j> =

af.

e

2

4.4. THEOREM. Let {M,e} be a neargroup and for each aEM, let aP:M+M be defined by S(aP) Sea for 8EM, then aPESM'

e(aP) = a(eP) =a and P:M+SM is (1-1). PROOF.

4.1.b ~aPis onto and 4.1.c ~aPis (1-1) so aPESM. 4.1.a ~ e(aP) = a(eP) =a.

From aP = SP~ e(aP) = e(SP) ~a 8 it fellows that

P

is (1-1).

4.5. THEOREM.

If

M is a set with eEM, P:M+SMis such that VaEM:a(eP) = e(aP) =a and e i s defined by ae8 = a(SP) then {M,e} is a neargroup.

PROOF.

a. aee = eea a. b. Since aPESM c. Since aPESM

VaEM, V8EM, :i[YEM:yea= 8. a(yp) = S(yp) ~ a = 8 so aey for aEM, SEM, yEM.

BeY ~ a = 6

4.6. THEOREM. If G is an s~extension of A with r then {r,e} with aeS

=

aS for aEr and SEf is a neargroup.

PROOF. If n is the natural representation of G on r then anESr for aEf and a(en) = e(an) = a so according to 4.5. {r,e} is a ne·argroup.

4.7. DEFINITION. By an s-extension of A with a neargroup {r,e} is to be understood an S-extension of a group A with the set r where as = aeS for aEr and 8Er. In this case the near-group should be considered as given.

Five

Equivalent S-extensions

Let G be an S-extension of A with

r.

For aEf and sEAUf are

a _{s and} s

a the S-functions belonging to G.

Let f:r~A be a mapp of into A with ef e. 1:G+G is defined by aa1 a(af)a. for aEA and aEf. 1 is (1-1) and onto and ac =a for aEA, aa:-:-1 A new operatien on the set G, which is indicated fined by: _{gl """gz}

₌

gl 1 ( T) T-l for _gl EG and _g, EG,

by * is

de-is isomorphism

.,..

T an of G onto G where a* is the structure in

which G is the set and * the operation. For

g₁ *-g

2 T "' g1 T ( r),

It is evident that for aEA, bEA, o.Er and SE

a*a a a,

a*a af.aa.( )-'.aa,

Bf

a*B af • a ( !3 f) • a !3 • ( ( a B f) 13 f) -• • ( aS f ) S •

r is a system of representatives of the right cosets (s.r.~) of A in G* because:

A*f

=

Ar = G and a*a = b*S ~ aa = bS ~ a =

s.

The S-functions of

o*

with reference to the subgroup A and the s.r.r.

r

are called et*s and a.*5 so that

a... .._a l l * *a _{for and}

a*a = a~ a. a aE r aEA.

a*B U* B*a *S a* fl,a *B for aE f and BEf.

Fr om this it fellows, because a*sEA and a.

r

for o.Ef and sE:Aur that <x* af. aa. (a.af)-1

a *a a o; a

.

Sf af.a(Sf).a S.((o.Bf)Bf)-•. (al3f)S,

G and a* are called equivalent S-extensions of A with

r

by means of f:f+A or by means of ' ·

5.1. THEOREM.

Let G and G~ be equivalent S-extensions of A with r by means of <:G+G then Gis also an S-extension of A with fT and for aer and sEA

u

r.

aTo sT= a*s and aT0

sT

=

a.w-s, in which a<c s< and aT0

sT are the S-functions of G with respect to the group A and the s.r.r. fT.

PROOF. T is an isomorphism of a* onto G and a<

=

a for aeA. aT• _{sT • aT} o ST

=

_a ( ) a* ""S a*' ( *S )

1: s 1: = a*S 1:

=

S*<l < = s a 1:

Let ~:G+Sr be the natural representation of G on r. and

*

.

*

o:G +Sr be the natural representat1on of G on r. (a"*b)*fl = (ab)*S = (ab(Sf))S for bEA, aer and Ser. So a(ga) = a(gTTT) for aer and geG, and a = TTT•

5.2 REMARK. Two S-extensions of a group A with r can be isomorphic as a group without being equivalent as B-exten~ions. We may get this for instanee from two subgroups A and B of G, which are isomorphic, while A is normal and B is not.

A simple example fellows.

5.3. EXAMPLE. Let the dihedralgroup D

₂

n(neN,n~2) be gener-ated by s and t , s2n

=

1, t 2 = 1 and tst = s-1

•

The centre of D₂n is {e,sn}. So D

2n is isomorphic with .an

S-extension of C2 with the set r of the natural numbers <2n with 8-functions a• satisfying a• = _{a for aer and aec 2 •}

Put A = {e,t}. So A is a subgroup of D isomorphic with C 2•

2D k -k •

< s) as s.r.r. of A in D

2n can be taken. From s t

=

ts 1t follows D₂n is isomorphic with an S-extension of c

2 with r, where the S-function a*a satisfies a*a - -a (mod 2n) for aer, aec₂ and a# e. Because ~aer with a

t

-a (mod 2n), the two S-functions are different. So there are tvo S-extensions of C2 with

r

which are isomorphic as group and not equivalent as s-extension.

is

5.4. THEOREM. If G is an S-extension of A with

r

and if a

an element of

r,

for which Aa = {e},then there is an

s-extension

o*

which is equivalent to G and for which:

VaEA, :S*a

=

e,

A r . ;r(e,*a)

'ifaEA,VSEa , Vye, .y ( y *S )*a.

a b

PROOF.

=

{e} implies a a ~ a = b for all aEA, bEA. eEaA ~ e

=

a, But then Aa

=

{e} means A

=

{e}.

So the theerem is trivial.

Nov suppose e~aA. Therefore we may define f:r~A by

Bf = (ab)- 1 for S = a.b and bEA, Bf is an arbitrary element of

A

for S~aA and

S

i

e and ef = e.

Then for 8

b

0: ... a

o:b vith bEA and for aEA: (a ..-b

)..-a

(af.ab.(abf)-I )-1 .(o:f).a(ba).(abaf)-1 (ab. (ab)-1

)-I. 0

(ba). (a(ba) )-I

=

e and

.... ( s*"a)

y

5.5.

REMARK. For the above S-extensions Core(A) Core(A)CAét.

{e} becàuse

5.6. THEOREM. If G is an S-extension of A with r and if a is an element of

r

for which Aa

=

{e}, then there is an

s-extension G~, equivalent to G and for which:

VaEA, VSEaA: s~ a

=

a,

VaeA, VSEaA,

Vyer:(/'-s)~a

=

(y*a)*"(S*"B.).

PROOF. By the same arguments as in the pro of of

5.4.

we may define

r: r-+A by Sf

₌

b-I ( ab) for 13

=

a b and bEA, 13f is an arbitrary element of A for Stia A and

_s

_:f

e and ef e. Th en i t fellows that for s

=

a b with bEA and aEA:

,!l ... a ab* _{a =} ( _{af. b. a f} a ( b )-• )-1 ( _{• af •} ) a( _{ba • a} ) ( ba _f )-1

5.7. DEFINITION. An s-extension of A with

r

is called a split S-extension iff vaer, V!lEf:a!l = e.

5.8. THEOREM. An S-extension of A with

r

is equivalent to a split S-extension iff ~f:r+A with:

a a!lf !lf !l

VaEr, V!lEf:af. (!lf). !l =(a ) f. PROOF.

a*e

=

af.a(!lf). a!lfe.((a!lf)!lf)-•.

5.9. THEOREM. If

a

is an S-extension of A with

r

and

~f:r+A such that H

=

{afalaer} is a subgroup of a then for the S-extension a* which is equivalent to a by means of f vaer, vser:a*e = e and r is a subgroup of a*.

If ~oreover H is a normal subgroup of

a

then

VaEr, VaEA:a*a a.

In this case

r

is a normal subgroup of a* and is called a normal complement of A in

a ...

PROOF. T:G ... +G defined by aaT = a(af)a for aEA and aEr is an isomorhism for which fT

=

H (see the beginning of this section). Therefore H is a subgroup of a * r is a subgroup of G*. So VaEr, V!lEf:a ... !l*a ... !l = att-6Er or a*e

=

e.

H is a normal subgroup of G * r is a normal subgroup of a*. a* a

=

a.

5.10. THEOREM. If a is an s-extension of A with r,a* equi-valent to a, 4l: r+r defined by (a.p )a = e and f: r+r defined by

Si

x

A system of two-sided representatives

In this section $:r~r is the mapping defined by (a$)a e, so ct$ = ä for aer.

A system of representatives of the right cosets of A in G is written as s.r.r. of A in G and a system of representatives of the left cosets as s.l.r. With s.t.r. is meant a common system of representatives of the left and the right cosets of A in G.

In 6.1. it is proved that in an S-extension of A with r,r is a s.t.r. iff $ i s (1-1) and onto and in

6.7.

we shall prove that r is a s.t.r. iff VaEf the mapping a~aa is (1-1) and onto. Van der Waerden proved by using a combinatorical theorem that if

G

is finite every subgroup of

G

has a s.t.r. and Zassenhaus

[ 17] extended this proof to the case that [ G:A) is finite.

In 6.10. we shall prove the more general theorem that a sub-group A of G has a s.t.r. if [A : Core(A)) is finite. This theo-rem includes obviously the case that A itself is finite.

In the remaining part of this section we furnishes a group with a subgroup without a s.t.r.

6.1~

THEOREM. If G is an s-extension of A with

r,

then

r

is a s.t.r. of A in G iff

$

is (1-1) and onto.

PROOF. Let r be a s.t.r. of A in G.

r is a s.l.r. • {a-1 laer} is a s.r.r.

=-

{(aa)-1_{alaEr} is a} s.r.r. • {ä!aer} is a s.r.r. ~ f$

=

r because f$Cf is evident. So: ~:r+r is onto.

Furthermorela~

= B$ •

a"'

6 => (aa)a-1 = (lfB)Il-1

w

• a(äa)-1

=

_s(lfa)-1

=> a

=

B

for r is a s.l.r. So: $:

r+ r

is ( 1-1 ) •

Conversely, let ~:r+r be (1-1) and onto.

f is a s.r.r. ~ f~ is a s.r.r. ~ (ëra.a-1

/aEf} is a s.r.r. =>

=> {a ( ä a. ) - l 1 a.E f } is a s • 1 . r • => f is a s . 1 . r • So r is a s.t.r. of A in G.

The following example ves an affirmative answer to the questions "can ~ be (1- ) without being onto?" and "can ~ be onto without being ( 1-1 )?".

6.2.

EXAMPLE. R 1s the set of the real numbers and

R R\ { 0}.

1

Gis a group on RxR₁ ,with the operatien (a,b)(c,d) == (a+bc,bd). A is the subgroup {(O,allaER }. F i s t h e s e t o f all f:R +R •

I I I

VfEF, r == {(t(tf),tf)/tER }U{(0,1)} is a s.r.r. of A in G

be-'

cause ifaER

1,1fbER1 , , 3:cER1 :(a,b) = (O,c)(t(tf),tf) and

thesetandcare unique. (0,1) is the representative of A.

The following computation, for ~. can be made. (t(tr),tr)-1

= (-t,(tf)-1

A(-t(tf)(-t(trlrl,-t(tf)r).

Hence: (t(tf),tf)~ (-t(tf')(-t(tf)f),-t(tf)f) and

(O,l)<P (0,1).

o:R+r is defined by to (t(tf),tf) for and Oo ( 0. 1 ) • Obviously o is (1-1) and onto.

g:R+R is defined by tg = -t(tf) for tER and Og 0.

I

Hence to<j> = tgo for tER, so o<j>o-1

=

g.

It fellows that ~is onto iff g i s onto and ~is

(1-1)

iff

g is (

1-1).

By

a suitable choice of f i t is easy to make g

(1-1)

and not onto or onto and not ( 1-1).

Let CCA and DCB, then $IC is the restrietion of $:A+B to C and <PIC:C+D is the restrietion of $:A+D to C.

6.3. THEOREM. If G is an s-extension of A with

r

and a.o:A+A is defined by a(a.o) = aa for aer and aeA, then

~~ A A -A . t "ff . t

PROOF. Let

~~aA:aA+~A

be onto. So VaEA.

~bEA:a

8 = ab. For these a and b the following conclusions can be made:

aaabEA ~ aaabEA ~ a-1_aabEA ~ _aabEaA ~ ~cEA: _{a. Hence ao is} onto.

Conversely let ao be onto. So VaEA. :!IbEA:ab = a.

-a _ab

It fellows that: a = a = ~ a • Hence ~

I

a :a +a

A A -A

is onto.

6.4. THEOREM.

If

G

is an S-extension of A with r and ao is defined as in

6.3.

then q,laA:aA+aA is (1-1) iff ao is

(1-1).

PROOF. Let

~~aA:aA+aA

be (1-1) and aEA, bEAthen a

aa = ab ... aa(aa)-1

=

_ab(ab)-1 _{=> a(aa)-}1

=

_b(ab)-1 _{... a(CL aa)-}1 _.aa

b

= b(a ab)-t a. b => a a

=> a = b.

Hence ao is (1-1).

Conversely let ao be (1-1) and a a

b

a •

a a ~ b

Put a .a = p and a .a

=

q. So pEA and qeA.

b

a b

I

A A -A

·a => a = a hence $ a :a +a is (1-1).

6.5. THEOREM.

If G is an S-extension of A with r, then for

A+ A

aer,(atj>) $ = a .

PROOF. From 3.21. it follows that

aA$C(a~)A,

so Cl

Let

i=

aa

for aEA and aer, then

B

=

äa

= ~ a

_aa = A

From

ä

=a (3.21.) it follows that BEa , so B Hence (atj>)A$caA.

A+ A (a.p) p a •

6.6. THEOREM.

1. q,:r+r is (1-1) iff Vaer,q,laA:aA+aA is (1-1). 2. q,:r~r is onto iff VaEI',4>IaA:aA~~A is onto.

PROOF.

I A A -A

1. If 4>:I'+f is (1-1) then 'ifaEf,Ç;,a :a +a obviously (1-1).

"' ( )-1 (8(3)-1 A

Conversely. a "' B =>a

=

(3 =>a

=

B (3.20) =>BE a • So yaEr,.plaA:aA .... c;A is (1-1) => q,:r~r is (1-1).

I

A A -A

2. t:r+r is onto • YaEf,. a :a +a is onto. This follovs from 6.5.

Conversely,

a

=

a

a ; a making use of the fact that

tl

:aA+~A

is onto, it fellows that 3:!3EaA vith 13$

I

A A -A . .

So Yaer,t a :a +a 1s onto => .p:r+r 1s onto.

6.7. COROLLARY.

=

Ct

1. That r i s a s.t.r. of A in G iff yaEr,.plaA:aA_,.;-A is (1-1) and onto fellows from 6.1. and 6.6.

2. That r i s a s.t.r. of A in G iff yaer,acr is (1-1) and onto (acr is defined as in 6.3.) fellows from 6.3., 6.4. and 6.7.1.

6.8. THEOREM. If G is an s-extension of A with

r

then

ao*

equivalent to G and having r as a s.t.r. of A in

o*

iff

\f aE r : o ( a A ) = o

(a

A ) •

a.

PROOF. Let

o*

be equivalent to G and r a s.t.r. of A in

o*'.

r:r+r is defined by (af)*<X

=

e.

6

*I

*A *"A ""' *A .

From .• 6. it follovs that 4> a :a ->-(atjl) lS ( 1-1) and onto.

·

*I

*A . *A A. ( ) .,.. ('· )A (: )

S1nce 4> a lS onto, a =a sectien

5 ,

and a<jJE atjl 5.10 it follovs that

aAr = (a*A>t

=

(af)*A = (af)A

=

(a.p)A.

Hence, because f i s

(1-1),

YaEf:o(aA)

=

o(aA).

. A A

Conversely let Vaer:o(a )

=

o(a }. Then 3:~:r+r vith wlaA:aA_,.~A is (1-1}

-a -b

Suppose a~

=

Bw => 3:aEA, S:bEA:a = (3

a.nd onto.

-~aaeA~

:B:bEA:äa ëb -=aa. =Bb => :B:a.EA, S"bEA:a = B => (a )-1 (ia)-1Bb aaEA, :B:bEA:a a

B

~

~:r+r is onto because ~E

Because

~~E~A

it is possible to take for every

~Er

an element

< ~n -~

~fEA such that ~~ = a • Because e~

=

e, we can choise ef = e. So f becomes a mapping from r into A.

Let

a*

be the S-extension of A with

r

equivalent to G by means of f,

*

.

(af)-1

From 5.10. it follows ~$

=

(a~) a~. So

*

<P

From

6.1.

it follows that r i s a s.t.r. of A in

o*

because ~ is (1-1} and onto.

6.9.

REMARKS.

1. o ( a A) • o (äA)

* [

A 11 : (A a )1r}

This follows from 3.18 •.

A-[ A:

al

2. o(aA)

=

o(~A)

*

The set of r cosets of A in AaA has the same order as the set of right cosets of A in A~A.

This follows from 3.15.

3. o(aA)

=

o(~A)

• The set of right cosets of A in

A~A

has the same order as the set of left cosets of A in AaA.

6.10. THEOREM. Let G be a group and A a subgroup of G.

If

[A:Core(A)]

is finite, a

s~t.r.

of

A

in

G

exists.

PROOF. This fellows from 3.20. and

6.8.

6.11.

THEOREM. If A is a finite subgroup of G, a s.t.r. of

A

in

G

exists.

PROOF. This follows from

6.10.

6.12. THEOREM. If A is a subgroup of G with finite index,

a s.t.r.

of

A

in G exists.

PROOF. This fellows trom 3.21. and

6.8.

6.13. THEOREM. If G is a Frobenius extension of A with

r,

an s-extension

G*

equivalent to

G

exists, having

r

as a s.t.r.

of A in G.