the group of units of centralizer near-rings

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The group of units of centralizer near-ringsJ.R. Clay a , C.J. Maxson b & J.D.P. Meldrum ca Dept. of Mathematics , University of Arizona , Tucson, Arizona, 85721, U.S.Ab Dept. of Mathematics , Texas A. & M. Univ , College Station, Texas, 77843, U.S.Ac Dept, of Mathematics , King’s Buildings, Mayfield Road, Edinburgh, EH9 3JZ, ScotlandPublished online: 27 Jun 2007.

To cite this article: J.R. Clay , C.J. Maxson & J.D.P. Meldrum (1984) The group of units of centralizer near-rings,Communications in Algebra, 12:21, 2591-2618, DOI: 10.1080/00927878408823122

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COMMUNICATIONS I N ALGEBRA, 12 (?.I), 2591-26 18 (1984)

THE GROUP OF UNITS OF CENTRALIZER NEAR-RINGS

J . R . Clay C. J. Maxson J.D.P. Meldrum

Dept. of Mathematics, Dept. of Mathematics, Dept. of Mathematics, Un ive r s i ty of Arizona, Texas A . & M . Univ. , King's Bu i ld ings , Tucson, Col lege S t a t i o n , Mayfield Road, Arizona 85721, Texas 77843, Edinburgh EH9 352, U.S.A. U.S.A. Scot land.

Let G be a group w r i t t e n a d d i t i v e l y , bu t no t n e c e s s a r i l y a b e l i a n

and l e t A be a group of automorphisms o f G w r i t t e n m u l t i p l i -

c a t i v e l y a c t i n g on t h e l e f t of G . Denote by C(A;G) t h e s e t of

a l l mappings of G i n t o i t s e l f which commute wi th A and map t h e

i d e n t i t y t o i t s e l f , t h e mappings a c t i n g on t h e r i g h t of G .

Hence

C(A;G) = {f : G + G ; (ax ) f = a ( x f J , a E k , x E G , O f = 0 } . G G

Then C(A;G) i s a nea r - r ing under pointwise a d d i t i o n of func t ions

and composit ion of f u n c t i o n s . That i s (C(A;G),+) is a group,

(not n e c e s s a r i l y a b e l i a n ) (C(A;G),.) i s a semigroup and

f (g+h) = f g + f h f o r a l l f , g , h i n C(A;G) . See P i l z [6] f o r -

b a s i c d e f i n i t i o n s and r e s u l t s a l though he uses t h e r i g h t d i s -

t r i b u t i v e law i n s t e a d of t h e l e f t d i s t r i b u t i v e law t h a t we u s e .

Moreover o u r nea r - r ings a r e a l l zero-symmetric, i . e . Of = 0 = fO,

where 0 i s t h e a d d i t i v e i d e n t i t y of C(A;G) . In t h i s paper we

a r e concerned wi th i d e n t i f y i n g t h e group of u n i t s of C(A;G) and

Copyright @ 1984 by Marcel Dekker, Inc.

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2592 CLAY, MAXSON, AND MELDRUM

establishing the existence of a determinant-like function on this ,

group of units.

In paragraph 1 we prove some group theoretical results which

are needed later on. Paragraph 2 introduces the group of units.

The next section deals with the determinant-like function. Our

final section gives an application.

1. Some group theory

For a set X , possibly infinite, we denote the symmetric

group on X by SymX or EX . Let H be any group and K a

subgroup of SymX. Construct H' , the set of all functions

from X to H . This is made into a group by defining the product as

fg(x) = f (x) . g(x) . What we now have is the Cartesian product of copies of H

indexed by the elements of X . We define K as a group of

automorphisms of H' by letting it permute the factors of HX

in the natural way. Formally

k f (x) = f(xk-l)

where f E H' , k E K , x E X . The wreath product of H by K

is the semidirect product of HA by K determined by this

X definition of K as a group of automorphisms of H , denoted

X H Wr K . Its elements are pairs {(k,f) ; k E K , f E H } with

the product given by

Technically this is the complete or unrestricted permutational

wreath product of H by K . See Scott [ I ] for further details

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CENTRALIZER NEAR-RINGS 2593

I t i s easy t o s e e t h a t t h e correspondence f + ( e , f ) , f o r

f E HX , e t h e i d e n t i t y of K , is an embedding of H' i n

H W r K , and t h e correspondence k + ( k , e ) , f o r k E K , e

t h e i d e n t i t y of HX is an embedding of K i n H W r K . I f

we i d e n t i f y HX and K w i t h t h e subgroups of H W r K t o which

they a r e isomorphic, we n o t i c e t h e fo l lowing: HX 4 H W r K ( a

means is a normal subgroup o f ) , K 5 H W r K , H W r K = H ~ K , X

and H n K = { e l . I n gene ra l t h i s means t h a t (k , f ) can be

k -1 w r i t t e n a s kf and t h a t f k = k f , s i n c e k f k = f k . This

is j u s t s ay ing t h a t when we " t r a n s f e r " t h e automorphism which

k E K induces i n H' t o t h e copy i n H W r K , it becomes t h e

r e s t r i c t i o n t o H~ of t h e i n n e r automorphism induced by K . Our f i r s t r e s u l t s concern de r ived groups. For an a r b i t r a r y

group M , t h e de r ived group is denoted M ' and

M' = ~ p ( [ m ~ , m ~ ] ; ml,m2 E M)

-1 -1 where [ml,m2] = m m2 mlm2 . Note t h a t M' q M and M/N i s

a b e l i a n i f and only i f N _> M' where N a M . We w r i t e

M M f = [M,M] . I f M = Gp (Y) then M y = G~ (rY1, y2] ; y1,y2 E Y) ,

i . e . t h e de r ived group is t h e normal c l o s u r e of t h e subgroup

genera ted by where y1,y2 run through a s e t of

gene ra to r s of M . We w r i t e M" f o r t h e der ived group of M ' . A t t h e o t h e r end of t h e spectrum, we d e f i n e t h e c e n t r e of M,Z(M),

by

Z(M) = {m E M ; mx = xm f o r a l l x E M} . Again Z(M) 0 M , and i s obviously a b e l i a n .

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We now r e t u r n t o wreath products , H W r K . Consider

Hx = { f E HX ; f ( y ) = e f o r a l l Y E X\{x}} , f o r some d i s -

t ingu i shed element x E X . Clear ly H i s a subgroup of H'

and f -t f ( x ) is an isomorphism from Hx t o H . I f f E Hx

and f ( x ) = h , we can w r i t e f a s hx . In t h i s case , i f

k - k E K , hx - hxk . In p a r t i c u l a r i f X i s f i n i t e , say

X = 1 , 2 , n , then f E H' can be w r i t t e n f = h"), . . h (n ) 1 n '

where f ( i ) = h( i ) . We need one more d e f i n i t i o n be fo re our

f i r s t r e s u l t . Let X = { 1 , 2 , ..., n) and l e t L be a subgroup

of H . Then de f ine

A ( L ) = i f E H' ; f ( l ) ... f ( n ) E LI

I t i s a r o u t i n e m a t t e r t o v e r i f y t h a t A(L) is a normal subgroup

of H' i f L 2 H I . Lemma 1.1. Let H be a group, X = { 1 , 2 , . . . , n} , K = S Y ~ X . Then

Proof . This proof p a r a l l e l s t h e work of P.M. Neumann [ 5 ] on - t h e s t andard wreath product , a s p e c i a l case of t h e permutat ional

wreath product .

We pick a s a s e t Y of genera to r s of H W r K t h e s e t H u K 1

From t h e remarks above t h i s is obviously a generat ing s e t Y f o r

H W r K . So (H W r K)' = G P ( [ Y ~ , Y ~ ] ; y1,y2 E Y) and thus

(H W r K)' _3H'K1 . Now (H W r K) ' a H W r K and (Hi)* = H ' 1 1 k

f o r a l l k E K . A s K i s t r a n s i t i v e on X , i t follows t h a t

H.' C (H W r K)' f o r 1 6 i 6 n . Hence (H W r K)' _ > H ' H ' . . . H A K ' . 1 - 1 2

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CENTRALIZER NEAR-RINGS

Now l e t f E A(H') . Then f ( 1 ) . . . f ( n ) = h E H 1 . So

-1 f 1 = f h s a t i s f i e s f ' ( 1 ) . . . f ' ( n ) = f ( 1 ) . . . f (n) h-I = e .

1

Thi s shows t h a t A(HV) 5 A({e}) H i . . . H ' . A s t h e r e v e r s e n

i n c l u s i o n is obvious , we have A(H1) = A({~})H; . . . H; . Now

( i ) -1 ( i ) ,k i ] = (hl hlki . Choose ki such t h a t l k . = i .

Then [ f ( i ) ( i ) -1 ( i ) , k i l = (hl ) hi . So

Hence f E (H W r K) ' and so A({e}) C (H W r K)' . This

enab le s us t o say A (H ' )K ' - C (H W r K) ' . We have shown t h a t i f

- 1 h . h . E X f o r a l l j , 1 5 i , j n , t hen A({e}) C X f o r 1 J

any subgroup X . We now need t h e r e v e r s e i n c l u s i o n . To do t h i s we show t h a t

A(H')K1 is a normal subgroup of H W r K con ta in ing a l l e lements

o f t h e form [ y , y ] w i th y . E Y = HI u K and use t h e remark 1 2

a t t h e beginning of t h e p roof . I f yl and y2 both come from K

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o r both from H1 , then [y1,y2] obviously l i e s i n A(H1)K' .

Since [y2,yl] = [ ~ ~ , y ~ ] - l , we only need t o cons ide r

y1 E HI , y E K t o cover a l l cases . Let yl = hl,y2 = k . 2

Then [hl ,k] = h-'hk = h-'h and t h i s obviously l i e s i n 1 1 1 l k

~ ( { e ) ) 5 A(H') . Hence A(H')K1 3 {[Y1,y2] ; yiE Y) . A l l

t h a t is l e f t is t o show t h a t A(H1)K' 0 H W r K . Note t h a t

X A ( H . ) a H , s i n c e A ( H ? ) 2 H;. . . H I = (H')* . ~ e t

n

f EA(H1) , k E K . Let f k = f 1 say. Then

-1 -1 - f ' (1 ) . . . f ' (n) = f ( l k ) . . . f (nk ) = f (1 ) . . . f (n) mod H ' s i n c e

- 1 k is a permutation of 1 , n . Since f ( 1 ) . . . f ( n ) E H '

s o does 1 f ( n , i . . f ' E A(H7) . Thus A(H1) is

X normalised by H' and K , so i s normal i n H K = H W r K .

-k Let k E K ' , f E H' . Then kf = f - lk f = k f f = k f * say.

Now f * ( i ) = f- ' ( ik-l) f ( i ) s o f * ( l ) . . . f*(n) =

-1 -1 -1 -1 f ( l k ) f l f (nk ) f ( n ) 2 e mod H ' s i n c e by rea r rang ing

t h e o r d e r of t h e f a c t o r s we get e . So f* E A(Ht) and

kf E K1b(H') = A(H1)K' a s A(HT) a N Wr K . A l l t h e conjugates

of elements of K ' by elements of H' l i e i n A(H')K1 . So

do a l l conjugates of elements of K ' by elements of K . Hence

A(H7)K' H W r K . This f i n i s h e s t h e proof .

Note t h a t a l l we r e a l l y need i s t h a t K i s a t r a n s i t i v e per-

mutation group on X . But we only use t h e r e s u l t when

K = Sym X .

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Lemma 1 . 2 . Let H be a group X = { 1 , 2 , ..., n}, n 2 5 ,

K = Sym X . Then

(H W r K)" = (H W r K)' = A(H')K1 . Proof . I t is only t h e f i r s t e q u a l i t y t h a t needs t o be proved.

We w i l l show t h a t A(H1)K' 5 (H W r K)" which i s s u f f i c i e n t ,

s i n c e (H W r K)" - C (H W r K) ' . Since K t i s t h e a l t e r n a t i n g

group on n symbols and n > 5 , K ' i s non-abelian s imple .

Hence K" = K ' and K ' 5 (A(H1)K') ' = (H W r K)" . Note t h a t K '

i s t r a n s i t i v e on f l , ..., n} . Let i j j be e lements o f X ,

and l e t i k = j . Let R E K ' , i R = j , jR = i . Since n b 5 ,

aga in we can f i n d such an R i n K ' . Then

-1 - 1 -1 [hihi*, a] = h , h , h h = h:' h .h .h-' = hi2h2 . s i n c e

1 j i R j R 1 J J ~ j

-1 - 1 -2 2 hihik = [hi ,k] , i t fol lows t h a t hi h . E (H W r K)" f o r a l l

J

h € H , i j E 1 . n . Now choose d d i f f e r e n t from i , j . Let m E K ' s a t i s f y i m = d , and p E K' such t h a t

i p = d , dp = j . Since K ' i s t h e a l t e r n a t i n g group on X , t h i s

i s always p o s s i b l e . Then h h-',p] = h i l h . h h-I = [ i i m i m i p imp

-1 2 -1 h . h h E (H W r K)" . Then 1 d j

By t h e proof of lemma 1.1, i t fol lows t h a t A({e}) 5 (H W r K)" . Let h(1) ,h(2) E H . Then D

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Hence k h ( l ) ) - l h : l ) , h i 2 ) E (H W r K ) I 1 , s i n c e (H W r K) a H Sfr K . 1 = [(hi1))-. , h i2 ) ] (H W r K ) . Hence H : 1 - c (H W r K) and a s

(H W r K)" 4 H W r K , H; . . . H I C (H W r K)" . n -

Thus A({~})H;. . . H I K ' C (H W r K ) " , and a s A(H1) = A ( { ~ } ) H ; . . . H I n - n

(proof of lemma 1 . 1 ) , we deduce t h a t A(H')K' - C (H W r K)" . This

f i n i s h e s t h e p roof .

Note t h a t i f n 6 4 , t hen K " C K t and t h e r e s u l t of lemma 1.2

holds no longer .

We now c h a r a c t e r i s e t h e normal c l o s u r e of K i n H W r K .

Then

KH W r K = A(Ht)K . Proof . Let L denote

KH W r K . Then L i s t h e group genera ted

by a l l conjugates of elements of K by elements of H W r K . X

Now H W r K = H K = KH' s i n c e H' a H W r K . So a t y p i c a l

gene ra to r of L is kf , where k ,k E K , f E H' . But kl 1

k kl E K . So

f X L = ~ p ( k ~ , k ; k l , k E K , f E H )

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Let h . E Hi . Then [k ,h i ] = h y l h E L . Hence by t h e l a s t 1 lk 1

p a r t of t h e f i r s t h a l f of t h e proof of lemma 1.1, i t fol lows

t h a t A({e}) 5 L , s i n c e K is t r a n s i t i v e on X . The l a s t

p a r t of t h e proof of lemma 1 . 2 showed t h a t H I C (H W r K)". . 1 -

This only used t h e f a c t s t h a t f o r a l l i f j i n

~ , h ~ ' h . E (H W r K)" and (H W r K)" Q H W r K . Since K i s 1 3

-1 t r a n s i t i v e on X , h . h . E L and L a H W r K , t h e same method

lk 1

shows t h a t H: C L . Hence A ( { ~ ] ) H ; ... H ' = A(H') 5 L . We 1 - n

hi have t h a t L 2 A(H1)K . Conversely k = k[k,hi ] E KA(H1) =

A(H1)K , s i n c e [ k , h . ] E A(H') by t h e proof of lemma 1.1, and

A(Ht) a H W r K . Hence L 5 A(Ht)K . This f i n i s h e s t h e p roof .

The l a s t two lemmas enab le us t o d e s c r i b e L' where

L = K H W r K .

Lemma 1 . 4 . Let H be a group, X = { 1 , 2 , . . . , n} , n 2 5 ,

K = SymX, L = K Wr . Then

and

I L : ~ ' 1 = 2 . Proof. By lemmas 1.1 and 1 .3 ,

(H W r K)' = A(H1)K' 5 L = A(H')K 5 H W r K . A s (H W r K)" = (H W r K)' it fol lows t h a t L' = A(H1)K' and

L' = L" . The f i n a l p a r t fo l lows s i n c e K ' i s t h e a l t e r n a t i n g

group on n symbols and s o I K : K ' I = 2 . Dow

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The f i n a l r e s u l t i n t h i s s e c t i o n concerns t h e c e n t r e of H W r K . We use Z(G) t o denote t h e c e n t r e of G . We a l s o need t h e

c e n t r a l d iagonal subgroup D of H W r K def ined by

X D = ( f E H ; f ( i ) = f ( j ) f o r a l l i , j E X , f ( i ) E Z(H)) .

Lemma 1 . 5 . Let H be a group, X a s e t , K = Sym X . Then

Z(H W r K) = D . Proof . Let kf E Z(H W r K) where k E K , f E H' . Assume

t h a t k # e . Choose i E X such t h a t i k f i and l e t

h E H , h f e . Then

k kfhi = h . k f = k h . f = kh f .

i k

This f o r c e s fhi = hikf . Take t h e "i" component on both s i d e s :

f ( i ) h = f ( i ) , s i n c e i k # i . Hence h = e , a c o n t r a d i c t i o n .

X So we must have k = e and Z(H W r K) C H . Let

f E Z(H W r K) and assume t h a t t h e r e e x i s t s i j i n X such

t h a t f ( i ) # f ( j ) . Choose k E K such t h a t i k - I = j . A s

f E Z(H W r K) , it fol lows t h a t f k = f , and hence

k f ( i ) = f (i) = f ( i k - l ) = f ( j ) , a c o n t r a d i c t i o n . So we must

have f (i) = f ( j ) f o r a l l i , j . F i n a l l y

X X Z(H W r K) - C Z(H = Z(H) by a s t andard r e s u l t from group theory .

Thus Z(H W r K) 5 D . Conversely l e t f E D , f ' E H' . Then f f ' ( i ) = f ( i ) f t ( i ) =

f f = f f f o r a l l i E X , s i n c e f ( i ) E Z(H) . So

every element of D commutes wi th every element of H' . Let

k -1 k E K . Then f ( i ) = f ( i k ) = f ( i ) f o r a l l i E X , a s f E D .

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So f k = f f o r k E K and hence every element of D commutes

X with every element of K . Thus D 5 Z(H K) = Z ( H W r K) . This

proves t h e r e s u l t .

2 . The group of u n i t s

We s t a r t w i th a key r e s u l t due t o Betsch [A] concerning t h e

e x i s t e n c e of maps i n C(A;G) , A proof of i t can be found i n

Maxson and Smith [z] o r Meldrum and Oswald [z]. I n o r d e r t o

s t a t e t h i s we need t h e fo l lowing d e f i n i t i o n . We denote by S t ( x )

t h e s t a b i l i s e r i n A of an element x of G , t h a t is

S t ( x ) = { a € A ; ax = x) . The fol lowing r e s u l t i s d u e t o B e t s c h [ l ] . -

Lemma 2 .1 . Let x ,y E G . There is an element f E C(A;G)

which s a t i s f i e s xf = y i f and only i f S t ( x ) C S t ( y ) . We denote t h e group of u n i t s of C(A ; G ) by U . An immediate

consequence of Be t sch ' s Lemma is t h e fo l lowing c o r o l l a r y .

Coro l l a ry 2 . 2 . Let f € U , x , y E G . I f xf = y , then

S t ( x ) = S t ( y ) . We s t a t e two more r e s u l t s about s t a b i l i z e r s which a r e wel l known,

and a r e s t a t e d h e r e a s they form a key p a r t of t h e sequence of

r e s u l t s .

Lemma 2.3 . Let x E G , a E A . Then

S t ( a x ) = aS t (x ) a-' . Coro l l a ry 2 .4 . The s e t of s t a b i l i z e r s o f t h e elements of an

o r b i t of A on G forms a conjugacy c l a s s of subgroups of G .

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Hence we speak of t h e conjugacy c l a s s of: s t a b i l i z e r s

a s s o c i a t e d wi th an o r b i t of A on G . D e f i n i t i o n 2 . 5 . We say t h a t two o r b i t s Ax,Ay of A on G a r e

synonymous i f t h e same conjugacy c l a s s of s t a b i l i z e r s i s

a s s o c i a t e d wi th each o r b i t .

The fo l lowing r e s u l t is r o u t i n e .

Lemma 2.6 . The r e l a t i o n "synonymous" i s an equivalence r e l a t i o n .

From t h e d e f i n i t i o n of C(A ; G ) it fol lows t h a t f E C(A ; G)

maps o r b i t s of A on G t o o r b i t s : (Ax)f = A(xf) . This is

t h e background f o r t h e next r e s u l t

Lemma 2 .7 . Let f E U . Then (Ax)f i s synonymous t o Ax . Proof . We use c o r o l l a r y 2.2, lemma 2.3 and c o r o l l a r y 2.4.

Since S t ( x ) = S t ( x f ) t h e conjugacy c l a s s a s s o c i a t e d w i t h Ax

i s t h e same a s t h a t a s s o c i a t e d wi th A(xf) = (Ax)f . Coro l l a ry 2 .8 . Let f E U . Then f permutes t h e o r b i t s i n a

given equivalence c l a s s of synonymous o r b i t s amongst themselves.

Let {Ej ; j E J) be t h e s e t of equivalence c l a s s e s of o r b i t s .

Let E = {Oij ; i E I . I be t h e s e t of o r b i t s i n E j '

Choose j J

a r e p r e s e n t a t i v e x E 0 . . f o r each o r b i t . Without l o s s of i j IJ

g e n e r a l i t y , we may assume t h a t S t ( x . . ) = S t ( x ) f o r a l l 1 J k j

i,k E I j *

We use f o r t h i s lemma 2.3 and t h e d e f i n i t i o n of t h e

equivalence c l a s s e s . Denote S t ( x i j ) by S j accordingly .

Define H = {x E I)O . S t ( x ) = s . I . Let j x i j ' J

-1 N ( s . ) = {b E A ; b S.b = S . 1 . I f u E H , a € N (S . ) , then

A J J J j A J

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-1 S t (au) = a S t ( u ) a = S t ( u ) . So we have most of t h e fo l lowing

r e s u l t .

Lemma 2 . 9 . The elements of N (S ) / S j a c t f i x e d p o i n t f r e e on A 5

t h e elements of H j '

Proof . We have a l r eady shown t h a t t h e elements of N (S ) A j

permute t h e elements of H . Since S t ( u ) = S f o r a l l j j

u E H t h e r e s t fo l lows immediately. j '

- NA(Sj)/Sj I f a E N (S ) we denote t h e image of a i n

A j by a

and then f o r u E H . J '

Lemma 2.10. Let f E U

P roof . J u s t apply c o r o l

- we w r i t e au = au .

. Then H.f = H f o r a l l j E J . J j

l a r y 2 . 2 , c o r o l l a r y 2 . 8 and t h e

d e f i n i t i o n of H These show t h a t H . f C H A s f has an j J - j '

i n v e r s e i n U , i t must be a permutat ion when r e s t r i c t e d t o H j '

Let B denote t h e s e t of permutat ions of H which commute j j

wi th t h e a c t i o n of N (S .) , i . e . A J

B. = I f : H , -+ H ; f a b i j e c t i o n and (au) f = h u f ) f o r a l l J J j

E N (S ) / S j and u E H . ) . A j J

Then B is a m u l t i p l i c a t i v e group and s o i s II B t h e j jEJ j '

complete o r Car t e s i an product of B , j E 3 j

Theorem 2.11. U i s isomorphic t o II B jEJ j

Proof . We remark t h a t f E C(A ; G ) i s uniquely de f ined by t h e

images of a s e t of o r b i t r e p r e s e n t a t i v e s . Now d e f i n e

9 : u - + n B by f 0 = ( f j ) j e J where f i s t h e r e s t r i c t i o n jEJ j j

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CLAY, MAXSON, AND MELDRUM

of f to H . By lemma 2.10 and lemma 2.9, it follows that 8 j

does indeed map U to II B . That it is a group homomorphism J j

is obvious from the definition. Since u H contains a jEJ j

complete set of orbit representatives, the set Ifj ; j E J}

uniquely determines f . Hence 9 is a monomorphism.

It remains to show that 8 is a surjection. Let f E C(A;G) . Then f is completely determined by the images of a set of

orbit representatives, since (ax)f = a(xf) for all

is a map from

set {hjIjEJ

define a map

. By corollary 2.2, if f E U , then

determined uniquely by a set {hj} jEJ

{x. ; i E I to H . Conversely 1 j J j

of maps from {x. ; i E I. to H 1 j J j '

x. .f E H 1 J j '

where h j

given such a

we can

f E U by (ax. .)f = a(x h.) . To check that f 1 J ij J

and is in U is routine. Hence there is a 1 - 1 correspondence between the elements of U and {{hjljEJ ;hj is

a map from {xij ; i E I . ) to H . } . On the other hand, by a J J

similar argument, an element of B is uniquely determined by j

its action on the representatives of orbits of N (S ) on H A j j -1

If a E A and u E H j '

then St(au) =aSt(u)a andso

au E H if and only if j a E NA(Sj)

. Hence the orbits of

N (S ) on H. are just the intersection of the orbits of A A j J

with H. . Thus, as for f E U , the elements of B. are in J J

1 - 1 correspondence with maps from x . . ; i E I . to H 1 J J j

This is enough to prove that 8 is a surjection.

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W e conclude t h i s s e c t i o n by p r e s e n t i n g ano the r c h a r a c t e r i z a t i o n

of t h e groups B If we w r i t e A f o r N (S ) /S A j j ' t hen B j ' j j

i s t h e group of permutat ions of H . which commute wi th t h e J

f i x e d p o i n t f r e e a c t i o n of A . on H j

From t h e proof of J

theorem 2 .11 a

Theorem 2.12.

Proof . Let f

where

and a ( f ) ( i ) =

I j

s e t of o r b i t r e p r e s e n t a t i v e s of t h e o r b i t s o f

given by { x . . ; i € I.] , 1 J J

B is isomorphic t o A . W r Sym(1 .) . j J J

E B Then x . . f = a .x .1 f o r some j ' 1 J 1 1 j

I . . We map B t o A . W r Sym(1.) by t h e map 6 J j J J

f0 = a ( f ) ~ ( f ) ,

ai , ~ ( f ) E Sym(Ij) maps i t o i ' . C e r t a i n l y

a ( f ) E A j - Since f i s a permutation of H and maps o r b i t s j

t o o r b i t s , i t fol lows t h a t f permutes t h e o r b i t s . This induced

permutation i s p r e c i s e l y T ( f ) . So 0 maps B t o j

A . W r Sym(1.) . Note t h a t i n t h e express ion f o r f0 we w r i t e J J

lj t h e element of A f i r s t , t h e element of Sym(1.) a f t e r ,

j J

which is t h e o p p o s i t e of t h e usua l n o t a t i o n i n s e c t i o n 1. Since

we can choose t h e images of o r b i t r e p r e s e n t a t i v e s a r b i t r a r i l y ,

i t fo l lows t h a t 0 is a s u r j e c t i o n . Since f i s uniquely

determined by t h e images of t h e o r b i t r e p r e s e n t a t i v e s , 8 is an

i n j e c t i o n . To f i n i s h t h e proof we need t o show t h a t 6 is a

homomorphism.

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Let f , h E B , f9 = a ( f ) a ( f ) , h9 = a ( h ) ~ ( h ) . Then j

x ( f h ) = (aixi, j)h i j

- - a . (xi, j > h 1

t - - aiairXiwj

where a = a ( f ) ( i ) , a;, = a ( h ) ( i l ) , iw = i r a ( h ) , i

t h a t i s in = i ~ ( f ) a ( h ) . Hence

But

and

= a ( f ) ( i ) a ( h ) ( i r ) . Thus f0h0 = ( fh )8 and 0 i s a homomorphism.

Combining theorems 2 . 1 1 and 2 .12 we o b t a i n

Corol lary 2 .13. U is isomorphic t o

n (N (S ) / S . ) W r Sym(1.) . jEJ A j J J

Spec ia l Case 1. I f A a c t s f ixed po in t f r e e on G , then U

i s isomorphic t o A W r Sym(1) , where I i s an index s e t of

t h e o r b i t s of A on G . Spec ia l Case 2 . I f C(A;G) i s r e g u l a r , t h a t i s , i f a l l

s t a b i l i z e r s s a t i s f y S t ( x ) C S t ( y ) impl ies S t (x) = S t ( y )

(Meldrum and Oswald [?I, Z e l l e r [ g ] , Meldrum and Z e l l e r [s] ) , then t h e H . a r e subgroups and one f i n d s t h a t

.I

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where A and H are as defined above. Hence j j

3. A determinant like function

In this section we consider a function from C(A;G) to an

abelian group which will turn out to behave somewhat like a

determinant function does for rings of matrices. As might be

expected in such a case, certain finiteness conditions have to be

imposed before the function can be defined. In order to keep the

situation as general as possible, we obtain the desired finite-

ness by restricting the domain of definition of the function. We

use the notation developed in the later part of section 2.

Definition 3.1. An element f E C(A;G) is said to be amenable

if given any equivalence class E of synonymous orbits, there j

is only a finite set of orbits (0; ; 1 6 k < n} where n

depends on j , such that OLf E E j '

Let (Ab)j he the abelianized A i.e. the quotient group of j

A . by its derived group. We now define a function D from the J

amenable elements of C(A;G) to J(Ab)j u {o} , the complete

product of the (Ab)j together with 0 , a zero element. Dow

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D(f) = 0 if for some j E J , there is an 0 . . E E such that 1 5 j

Gf n Oij = g . D(f) = 0 if for some 1 6 k # k t $ n , 0' f = 0 ' f ,

k k'

D(f) = (aj)jEJ ,

where a. is defined as follows. If {O; ; 1 6 k 6 n} is the J

finite set of orbits mapped into E. by f , then J

a = ( II a(k))Aj , where x'f = a(k)~~(~) , xi is the j l<k<n

chosen representative of oi,Aj is the derived group of A. J

and a(k) E A j '

This function D is our determinant-like function. The most

obvious case in which it is defined on all of C(A;G) is when

there are only a finite number of orbits of A on G . We will

now prove that D has some determinant-like properties. First

we will show that its value is independent of the choice of

representatives {xij ; i E I. , j E J} which we will call in J

future a basis.

Theorem 3.2. Let f be an amenable element of C(A;G) . Then

the value of D(f) is independent of the choice of basis.

Proof. Let two bases be u . ; i € I. , j J and 1 J J

{vij ; i E I. j E J} and the corresponding values of D(f) J

be (aj)jEJ and (bj)jEJ respectively. Further

let v =c u , i E I. , j E J where c. E A . If for some ij ij ij J 1 j

j E J , there is an 0. . E E such that Gf n 0 . = p , then 1 3 j 1 j

D(f) = 0 irrespective of the choice of a basis. So there is

nothing to prove in this case.

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W e can t h e r e f o r e assume t h a t f o r each j E J , and each

C Gf . Since f is amenable t h i s f o r c e s each E i E I j , O i j - j

t o be f i n i t e . Again, t o exclude t h e t r i v i a l c a s e D(f) = 0 , we

may assume t h a t f induces a 1 - 1 map of t h e s e t of o r b i t s

onto themselves. Consider a p a i r of o r b i t s O1 and O 2 such

t h a t 0 f = 0 1 2 ' with r e p r e s e n t a t i v e s u1 ,u2 i n t h e o l d b a s i s ,

vl,v2 i n t h e new b a s i s vl = clul , v2 = c u 2 2 '

I f u f = au 1 2 '

v f = b v 1

, t hen c u f = bc u 1 1 . So clau2 = bc2u2 and

- 1 b E c1a(St(u2))c2 . The elements a (k ) i n t h e d e f i n i t i o n of

D(f) a r e only determined up t o a cose t of S . = S t ( x J i ( k ) j ) '

- 1 we may assume wi thout l o s s of g e n e r a l i t y t h a t b = c ac . The

1 2

element a i n t h e d e f i n i t i o n of D(f) i s a product of elements j

of A . / A ! , an a b e l i a n group. So we may reo rde r t h e product of J J

-1 t h e elements of t h e form b = c ac wi thout l o s s of

1 2

g e n e r a l i t y .

S ince f induces a 1 - 1 map from t h e s e t of non-zero o r b i t s

on to i t s e l f , every o r b i t occurs once a s "0 " and once a s "02". 1

So t h e corresponding "c and "c-l" a l l occur once. Using 1 2

commutativity, we s e e t h a t they cancel i n p a i r s , showing t h a t

D(f) i s uniquely de f ined .

We p o i n t ou t he re t h a t i f some E . i s i n f i n i t e , then by t h e J

d e f i n i t i o n of amenable and of t h e func t ion D , i t fol lows t h a t

D(f) = 0 f o r a l l amenable elements f . So t o avoid t r i v i a l i -

t i e s , we r e s t r i c t ou r se lves from now on t o t h e case t h a t E . i s J

f i n i t e f o r a l l j E J .

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Lemma 9.3. The funct ion D i s a s u r j e c t i o n .

Proof . Given ( a . ) E II (Ab) , we choose b . E A such J jEJ jEJ J j

t h a t b .A: = a and f i x i E I j . Define a m a p f by J J j

x f = b x and x f = x i l j

f o r a l l i E 1 i } . Then i j j i j i ' j J

D(f) = ( a . ) a s des i red . J jEJ

We now show t h a t D i s a homomorphism from U t o j:J(Ab) J .

Theorem 3.4. The func t ion D is a homomorphism from U t o

n ( ~ b ) . j€ J

Proof . Let f , f 7 E U and f i x j E J . A s E i s f i n i t e w e 3

w i l l w r i t e t h e o r b i t s i n E a s 01, . . . , 0, with r e p r e s e n t a t i v e s j

x 1 . . . , x . Note t h a t f and f ' permute these o r b i t s . Let n

a , a be t h e permutations of E l , . . . , n} such t h a t x f E Oi , i n

- - (a:)jEJ . We have x f ' E Oi . Let D(f) = ( a j ) jEJ , D(f l ) i Q

x f = a ( i ) x i ' f ' = a l ( i ) x Then

i a i '

x iaa f f l = ( a ( i a ) x i a ) f t = a ( i a ) a V ( i ) x So i

where b = ( n a ( i a ) a r ( i ) ) A ! = ( n a ( i j l ~ i ~ n i < i < n

This proves t h e r e s u l t .

Note t h a t we use t h e f a c t t h a t i s a permutation of { I , . . . , n}

and t h a t modulo A ' t h e elements of A commute. j j

We now r e s t r i c t our se lves t o cons ide r ing D a s a homomorphism

from U t o fl (Ab)j i n t h e case when a l l E a r e f i n i t e . We jEJ j

w i l l use N t o denote Ker D . This w i l l correspond t o t h e

ke rne l of t h e determinant func t ion , namely t h e s p e c i a l l i n e a r

group of ma t r i ces , i n t h e way t h a t U corresponds t o t h e general

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l i n e a r group of ma t r i ces . Using c o r o l l a r y 2.13 we w i l l w r i t e

U a s Jl A . W r Sym(Ij) , where A , = NA(Sj)/Sj . Let K jEJ J J j

denote Sym(Ij) . Write u E U a s

where a E K j , f . E A1' . From t h e d e f i n i t i o n of D , app l i ed j J J

t o t h e s p e c i a l case of t h e group of u n i t s , we o b t a i n t h e fol lowing

formula.

Lemma 3.5 . Let u = (a f ) be given a s above. Then j j jEJ

The proof of t h i s lemma is immediate us ing t h e d e f i n i t i o n s . We

can now i d e n t i f y N . Theorem 3.6. N = Ii A(Af)K

j E J j j '

Proof . From lemma 3 .5 , i t i s obvious t h a t A(A1)K C N f o r each j j -

j E J , and t h a t II A(A!)K. C N . On t h e o t h e r hand it i s a l s o jEJ J J -

obvious from t h e d e f i n i t i o n of a wreath product t h a t t h e map

obta ined by r e s t r i c t i n g t h e formula of lemma 3 .5 t o a f i x e d

A . W r K has a s ke rne l A(A;)Kj . This shows t h a t J 5

N = n A(A!)K s i n c e each f a c t o r can be t r e a t e d independent ly . jEJ J j

Corol lary 3 .7 . Let J be f i n i t e of o rde r n . Then N ' has

index 2n i n N and U 1 = N' = N f t . Proof . Apply lemmas 1 . 2 , 1 . 3 and 1 .4 t o t h e r e s u l t of theorem

Note h e r e t h e s l i g h t depar tu re from t h e corresponding behaviour

f o r matr ix groups, where GL(n,K) ' = SL(n,K)' = SL(n,X) except

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i n very s p e c i a l c a s e s .

We next look a t t h e c e n t r e of U

Theorem3.8 . Z(U) = II Z(A.WrK.) = I I D where D i s t h e jEJ J J j E J J j

c e n t r a l d iagonal subgroup of A . W r K. J j '

P roof . The proof fo l lows e a s i l y from lemma 1 .5 .

Since D . Z(A.) , we a l s o have t h e fo l lowing. J J

Coro l l a ry 3.9. Z(U) 2 .II Z(A,) . JEJ J

Theorem 3.10. Let n . be t h e ( f i n i t e ) number of e lements i n J

I Then j '

n Z(U) n N = { f E Z(U) ; f ( i ) j E A ' f o r each j E J) .

j

P roof . This fo l lows immediately from t h e previous r e s u l t s and

d e f i n i t i o n s .

We conclude t h i s s e c t i o n wi th a r e s u l t t h a t g ives a d i s t i n g u i s h e d

s e t of gene ra to r s f o r U and f o r N , genera to r s which resemble

somewhat t h e "nice" gene ra to r s of t h e corresponding groups of

m a t r i c e s . We w i l l d e f i n e t h r e e types of elements of C(A;G) . ( i ) For some j , and some p a i r i f i1 i n I and f o r some

j

-1 a E A j , xi j f = a x i l j , x i t j f = a x . and f f i x e s eve ry th ing

i j

o u t s i d e 0 . . u Oi, . 13

( i i ) For some j , some i E I and some a E A; , xi j f = ax j i j

and f f i x e s eve ry th ing o u t s i d e Oi j . ( i i i ) For some j , some i I and some a E A j , xijf = ax

j i j

and f f i x e s eve ry th ing o u t s i d e oij .

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Then t h e fol lowing is an easy deduction from s tandard group

t h e o r e t i c r e s u l t s and t h e e a r l i e r work,

Theorem 3.11. Let J be f i n i t e . Then t h e elements of types ( i )

and ( i i ) generate N , and t h e elements of types ( i ) and ( i i i )

generate U . Proof . Corol lary 2 .13 desc r ibes t h e s t r u c t u r e of U . Since any

permutation o f I . can be w r i t t e n a s a product of t r anspos i - J

t i o n s , a s u i t a b l e product o f elements of type ( i ) w i l l g ive us f

such t h a t f6 = v f o r any given TT E K . , using t h e n o t a t i o n J

of Theorem 2.12. Then elements of type ( i i i ) w i l l g ive us f

such t h a t 16 i s an a r b i t r a r y element of A K j . This covers j

t h e case of U . Theorem 3.6 can be used i n t h e same way t o

o b t a i n t h e r e s t of the theorem: elements of type ( i ) g ive us f

such t h a t f0 i s an a r b i t r a r y element of K o r A(ie3) while j

elements of type ( i i ) g ive us f such t h a t f6 i s an a r b i t r a r y

element of A; ... A ' using t h e n o t a t i o n of paragraph 1. n '

4 . Some a p p l i c a t i o n s

We now consider how some of t h e p r o p e r t i e s of t h e determinant of

matr ices t r a n s f e r t o t h e func t ion D def ined i n t h e l a s t sec t ion .

The f i r s t "matrix proper ty" we consider i s : d e t A = 0 i f and

only i f A i s not an i n v e r t i b l e ma t r ix , i . e . not a u n i t .

A s we remarked j u s t be fo re lemma 3 .3 , i f some E is i n f i n i t e , j

then D(f) = 0 f o r a l l amenable elements f , s o hencefor th we

assume the fol lowing.

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(4 .1 ) . Each equivalence c l a s s E of synonymous o r b i t s c o n s i s t s j

o f a f i n i t e number of o r b i t s .

We p o i n t ou t t h a t un les s (4 .1) ho lds , t h e i d e n t i t y map i n C(A;G)

i s not amenable, ano the r reason f o r making t h e assumption.

Lemma 4 .2 . Let f be amenable. Then D(f) # 0 i f and only i f

f induces a permutat ion of t h e s e t of o r b i t s .

P roof . Note t h a t by t h e d e f i n i t i o n of C(A;G) , f maps o r b i t s

t o o r b i t s . So f induces a map from t h e s e t of o r b i t s t o i t s e l f

The f i r s t p a r t of t h e d e f i n i t i o n shows t h a t t h i s induced map must

be s u r j e c t i v e i f D(f) # 0 . The second p a r t of t h e d e f i n i t i o n

shows t h a t i t must be one-one. So i f D(f) # 0 t hen f induces

a permutation of t h e s e t of o r b i t s .

Conversely, suppose t h a t f induces a permutation of t h e s e t of

o r b i t s . Then, i n t h i s case D(f) E II(Ab). a s can be seen from t h e j J

d e f i n i t i o n of D and s o D(f) # 0 . We have t h u s proved t h e

r e s u l t .

The fo l lowing ques t ion now a r i s e s . I s i t p o s s i b l e t o f i n d a p a i r

(A;G) which s a t i s f i e s ( 4 . 1 ) , and an element f E C(A;G) such

t h a t f induces a permutation on t h e s e t of o r b i t s bu t i s no t a

u n i t ? We con jec tu re t h e answer t o be yes . The fo l lowing

d e f i n i t i o n g ives an i n d i c a t i o n of where t o s t a r t looking.

D e f i n i t i o n 4 .3 . We say t h a t t h e p a i r (A;G) s a t i s f i e s t h e

f i n i t e n e s s cond i t ion ( f . c . ) i f given a E A , g E G then

-1 S t ( x ) g a-' S t ( x ) a impl i e s S t ( x ) = a S t ( x ) a . This d e f i n i t i o n was used by Z e l l e r [t3] . I t i s a s t r o n g e r

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cond i t ion than t h e one used i n Meldrum and Z e l l e r [ ? I , bea r ing

t h e same name.

Lemma 4.4 . Let (A;G) not s a t i s f y f . c . but s a t i s f y ( 4 . 1 ) . Then

t h e r e e x i s t s f E C(A;G) such t h a t D(f) f 0 and f i s not a

u n i t . Proof . Since ( A ; G ) does not s a t i s f y f . c . t h e r e e x i s t s a E A ,

g E G such t h a t S t ( x ) C as t (x)aml . Define f E C(A;G) by

xf = ax , and f i s t h e i d e n t i t y map on G \ { A X } . Since

(A;G) s a t i s f i e s ( 4 . 1 ) , f is amenable. Also f induces t h e

i d e n t i t y map on t h e s e t of o r b i t s . So D(f) f 0 . But

- 1 S t ( a x ) = a S t ( x ) a 3 S t ( x ) . Hence by c o r o l l a r y 2 .2 , f is

not a u n i t .

I f t h e index s e t J of t h e equivalence c l a s s e s of synonymous

o r b i t s i s i n f i n i t e t h e fol lowing s i t u a t i o n could a r i s e . Without

l o s s of g e n e r a l i t y assume t h a t J > Z , t h e s e t of i n t e g e r s . - Suppose t h a t 0; is an o r b i t i n En and t h a t r e p r e s e n t a t i v e s

X n of {OA}nEZ can be chosen s o t h a t S t ( x n ) C S ~ ( X ~ + ~ ) f o r

a l l n E Z . Then d e f i n e f E C(A;G) by x f = x ~ + ~ and f n

a c t s a s t h e i d e n t i t y on G \ u {Axn} . Then, a s i n lemma 4 . 4 , nEZ

D(f) f 0 but f is no t a u n i t

We now add t h e fol lowing assumption t o ( 4 . 1 ) , t o hold f o r t h e

r e s t of t h i s paper .

( 4 . 5 ) . The p a i r (A;G) s a t i s f i e s the f . c . and t h e s e t J of

equivalence c l a s s e s of synonymous o r b i t s i s f i n i t e .

In f a c t (4 .1) and (4 .5) a r e equivalent t o :

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(4 .6) t h e p a i r (A;G) s a t i s f i e s t h e f . c . and t h e r e a r e only a

f i n i t e number of o r b i t s .

We now show t h a t i f 4.6 i s s a t i s f i e d then D(f) # 0 i f and only

i f f i s a u n i t . F i r s t l e t us n o t e t h a t an element f E C(A;G)

i s a u n i t i f and only i f f i s a b i j e c t i o n . For i f f i s a

-1 b i j e c t i o n then f : G + G i s wel l -def ined. I t remains t o

- 1 show t h a t f E C(A;G) . Let y E G - 103 , a E A and l e t

yf-l = x , ( a y ) f - l = z . Then z f = ay = a ( x f ) = ( ax ) f s i n c e

f E C(A;G) . But then z = ax s i n c e f i s a b i j e c t i o n s o

- 1 (ay)f- ' = a(yf ) a s d e s i r e d .

Theorem 4 .7 . Under t h e assumption (4.6) f o r f E C(A;G) ,

D(f) # 0 i f and only i f f i s a u n i t .

P roof . From lemma 4.2 we know D(f) # 0 i f and only i f f

induces a permutat ion on t h e s e t of o r b i t s which is f i n i t e . So

given xl E G , we can f i n d x 2 , . . . , ~ i n G such t h a t m

x . f E , i = 1 , 2 , . . . , m - 1 and x f E Axl . Thus us ing m

B e t s c h ' s lemma we o b t a i n S t (x l ) 5 S t ( x l f ) 5 S t (ax l ) f o r some

a E A . From t h e f i n i t e n e s s cond i t ion t h i s g ives

S t ( x l ) = S t ( x l f ) . Since x was a r b i t r a r y S t ( x ) = S t ( x f ) f o r 1

a l l x E G . Now suppose D(f) f 0 . From t h e f i r s t p a r t of t h e

d e f i n i t i o n of D,Gf must i n t e r s e c t every o r b i t and consequen t ly f

must be s u r j e c t i v e . I f xf = yf then we must have Ax = Ay f o r

o the rwise D(f) = 0 . I f y = bx then xf = yf impl i e s

xf = bxf and s o b E S t ( x f ) = S t ( x ) . But then x = y and s o f

is a b i j e c t i o n and hence a u n i t . The converse is obvious .

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Recal l t h a t a p a i r (A;G) i s r e g u l a r i f f o r

x ,y E G - ( 0 ) , S t ( x ) C_ S t ( y ) impl ies S t ( x ) = S t ( y ) . Since

r e g u l a r i t y impl ies f . c . we have t h e fol lowing s p e c i a l c a s e of

theorem 4.7.

Coro l l a ry 4.8. Let (A;G) be r e g u l a r wi th a f i n i t e number of

o r b i t s . For f E C(A;G) , D(f) 9 0 i f and only i f f i s a u n i t .

We next look a t t h e a l t e r n a t i n g p roper ty of t h e ma t r ix determinant.

A s t h e " e n t r i e s " i n our case come from a group and not a r i n g , we

use t h e fol lowing form: i f a matr ix has two columns ( o r rows)

i d e n t i c a l , then i t s determinant i s ze ro .

Theorem 4 .9 . I f t h e element f E C(A;G) maps two d i s t i n c t

o r b i t s t o t h e same o r b i t , then D(f) = 0 . Proof . Di rec t from t h e d e f i n i t i o n of D , o r from lemma 4 . 4 ,

s i n c e a map on a f i n i t e s e t is a permutation i f and only i f i t is

1 - 1 .

Let f E C(A;G) , and suppose g E C(A;G) d i f f e r s from f only

i n t h e e f f e c t on one o r b i t , say Ax and t h a t axf = xg . Theorem 4.10. Let f , g i n C(A;G) be def ined a s above. Then

D(f) .aA1 = D(g) j

where (Ax) f E E j '

Proof. The r e s u l t i s immediate from t h e d e f i n i t i o n of D . Again, a s t h e " e n t r i e s " i n our case a r e from a group, t h i s is

t h e form t h e m u l t i l i n e a r p roper ty of t h e ma t r ix determinant t a k e s .

F i n a l l y we mention t h e concept of e igenvalues - e igenvec to r s .

Because o r b i t s of A r ep lace one-dimensional subspaces , and G

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i s t h e s e t - t h e o r e t i c union of i t s o r b i t s , an e igenvec to r of f

i s an element x of G such t h a t xf = ax . So t h e e x i s t e n c e

of an e igenvec to r f o r a given element of C(A;G) w i l l be obvious

from t h e d e f i n i t i o n of t h e e lement . I f t h e b a s i s i s n o t a b a s i s

of e igenvec to r s then t h e r e i s no p o s s i b i l i t y of changing t h e

b a s i s t o o b t a i n a b a s i s of e igenvec to r s .

References

[I] G . Betsch. Near-rings of group mappings. Oberwolfach

Conference 1976.

[2] C . J . Maxson, K . Smith. The c e n t r a l i z e r o f a s e t of group

automorphisms. Corn. i n Algebra 8 (1980), 211-230.

[3] J .D.P. Meldrum, A . Oswald. Near-rings of mappings. Proc.

Royal Soc. E d i n b u r g h , e A (1979), 213-223.

[4 ] J.D.P. Meldrum, M . Z e l l e r . The s i m p l i c i t y of nea r - r ings of

mappings. Proc. Royal Soc. Edinburgh. =A (1981), 185-193.

[5] P.M. Neumann. The s t r u c t u r e of s t andard wreath p roduc t s .

Math. Z. - 84 (1964) . 343-373.

[6] G . P i l z . Near-rings. North Holland/American E l s e v i e r .

Amsterdam, New York 1977.

[7 ] W.R. S c o t t . Group Theory. P r e n t i c e Hall.Englewood C l i f f s ,

New J e r s e y 1964.

[8] M . Z e l l e r . C e n t r a l i z e r near- r ings on i n f i n i t e groups.

Doctora l d i s s e r t a t i o n . Texas A . & M . Un ive r s i ty 1980.

Received: J u l y 1982

Revised: June 1984

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the group of units of centralizer near-rings

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