ON

FINITE

p-GROUPS,

SUBGROUPS A.

HAS D.

A

EACH

OF

METACYCLIC

WHOSE

PROPER

COMMUTANT

Ustyuzhaninov

UDC519.44

In t h i s p a p e r w e p r o v e t h e f o l l o w i n g t h e o r e m : T H E O R E M . S u p p o s e t h a t in t h e f i n i t e p - g r o u p G w i t h t w o - g e n e r a t o r c o m m u t a n t , e a c h p r o p e r s u b g r o u p i s m e t a c y c l i c . T h e n t h e c o m m u t a n t G ' of t h e g r o u p G h a s one of t h e f o r m s : 1) G ' i s a m e t a c y c l i c g r o u p ; 2) G ' i s a n e x t e n s i o n of a m e t a c y c l i c g r o u p by a c y c l i c g r o u p h a v i n g a t l e a s t t h r e e g e n e r a t o r s ; 3) G ' i s a n e l e m e n t a r y a b e l i a n g r o u p of o r d e r p4. Such a r e s u l t i s a l s o o b t a i n e d f o r f i n i t e p - g r o u p s w i t h a n a r b i t r a r y n u m b e r of g e n e r a t o r s , not proved here.

but it is

On t h e b a s i s of t h i s t h e o r e m w e c o m p l e t e l y d e t e r m i n e t h e s t r u c t u r e of the c o m m u t a n t s of f i n i t e p-. g r o u p s f o r w h i c h t h e c o m m u t a n t of e a c h p r o p e r s u b g r o u p i s c y c l i c . In p a r t i c u l a r , f o r p = 2, t h e c o m m u t a n t i s a b e l i a n w i t h a c y c l i c m a x i m a l s u b g r o u p . W e n o t e t h a t c o m m u t a t i v i t y of t h e c o m m u t a n t f o r odd p r i m e s p w a s p r o v e d e a r l i e r by A l p e r i n . N o t a t i o n : G = i s a g r o u p G g e n e r a t e d by t h e s e t of e l e m e n t s N. ~(G) d e n o t e s the F r a t t i n i s u b g r o u p of G. f2(HH) d e n o t e s the s u b g r o u p g e n e r a t e d by a l l e l e m e n t s of o r d e r p of t h e s u b g r o u p H. P(H) d e n o t e s t h e s u b g r o u p g e n e r a t e d by the p - t h p o w e r s of a l l t h e e l e m e n t s of t h e s u b g r o u p H. CG(f~) = C(N) d e n o t e s t h e c e n t r a l i z e r of t h e s u b s e t N in t h e g r o u p G. [x, y] = x - l y - ~ x y d e n o t e s t h e c o m m u t a t o r of t h e e l e mentsxandy. Ix1, . . . ,Xn] = [ [ x l , . . . , X n _ l ] X n ] , n > 2. [A, B] d e n o t e s t h e m u t u a l c o m m u t a n t of t h e s e t s A a n d B in t h e g r o u p G. G'=

[G,G],

G"=

[G',G'],

G ~ = [V',V].

B ( G ) ~ = G~r

I GI d e n o t e s the o r d e r of t h e g r o u p G, I(x>[ = Ix] d e n o t e s t h e o r d e r of t h e e l e m e n t x; 1G: HI d e n o t e s the i n d e x of t h e s u b g r o u p H in t h e g r o u p G. exp (H) d e n o t e s the e x p o n e n t of the s u b g r o u p H, i . e . , m a x { [ E l , h e H}. G = A X B d e n o t e s t h e s e m i d i r e c t p r o d u c t of t h e s u b g r o u p s A and B w i t h t h e i n v a r i a n t f a c t o r A. N A G d e n o t e s t h a t t h e s u b g r o u p N i s n o r m a l in the g r o u p G. d(G) = d d e n o t e s t h e m i n i m a 1 n u m b e r of g e n e r a t o r s of t h e g r o u p G, in p a r t i c u l a r , d ( ( l > ) = 0. or, cr1 d e n o t e e l e m e n t s of o r d e r p in G. r , ~1 d e n o t e e l e m e n t s of o r d e r p of Z(G) of t h e g r o u p G. a i . fib Yi, m i , ni, ki, l i , r i , s i, t i d e n o t e i n t e g e r s . Qn d e n o t e s t h e g e n e r a l i z e d g r o u p of q u a t e r n i o n s of o r d e r 2 n, n_> 3 ; Q =Q3. Dn d e n o t e s t h e d i h e d r a l g r o u p of o r d e r 2 n, n_> 3, D = D 3 o r , f o r p > 2, t h e n o n a b e l i a n g r o u p of o r d e r p3 a n d exp (D) = p. Vpn d e n o t e s the e l e m e n t a r y a b e l i a n g r o u p of o r d e r pn, n _> 2. A n (r, s ) g r o u p i s a f i n i t e p - g r o u p G in w h i c h t h e r e e x i s t s a n o n c y c l i c m e t a c y c l i c s u b g r o u p N s u c h t h a t G ' ~ N , r = d ( G ) , s = d ( G / N ) + 2. T h e c l a s s of f i n i t e p - g r o u p s , in w h i c h e a c h p r o p e r s u b g r o u p h a s a m e t a c y c t i c c o m m u t a n t w i l l b e d e n o t e d by t h e l e t t e r 9~. W e c o n s i d e r o n l y f i n i t e p - g r o u p s . W e s h a l l u s e s e v e r a l r e s u l t s w h i c h a r e p r o v e d in [1]. B I. The finite p-group B 2. If the finite p-group

G is metacyclic

if and only if G/B(G)

G is not metacyclic U=G/B(G)=(
is metacyclic.

and d(G) = 2, then X <~>) X,

T i l - - p ~,

T r a n s l a t e d f r o m S i b i r s k i i M a t e m a t i e h e s k i i Z h u r n a l , Vol. 12, No. 4, pp. 8 4 4 - 8 5 4 , ~ O r i g i n a l a r t i c l e s u b m i t t e d N o v e m b e r 10, 1969.

9 1972 Consultants Bureau, a division of Plenum Publishing Corporation, 227 ~/est 17th Street, New York, N. Y. 10011. All rights reserved. This article cannot be reproduced for any purpose whatsoever without permission of the publisher. A copy of this article is available from the publisher for $15.00.

1971.

1 607

a n d (m, n) i s the t y p e of the a b e l i a n g r o u p G / G ' . B~. If in t h e f i n i t e p - g r o u p G, d(G) = 2, then I G ' : B(G) [ = p and B(G) i s the o n l y s u b g r o u p of G' i n v a r i a n t in G h a v i n g t h i s p r o p e r t y . W e a l s o u s e a r e s u l t f r o m [2]: III. If in t h e f i n i t e p - g r o u p G t h e r e e x i s t s a s u b g r o u p H s u c h that d(H) : 2, H' ~ <1>, a n d [G, H] ~ H, t h e n G = HCG(H ).

I [G, H]I = p

F o r p r o o f of the t h e o r e m s o m e p r e l i m i n a r y r e s u l t s a r e n e c e s s a r y , w h i c h a r e p r o v e d in L e m m a s 1 - 5 and w h i c h a r e of i n t e r e s t in t h e m s e l v e s . W e n o t e t h a t L e m m a 3 g e n e r a l i z e s r e s u l t B~. L E M M A 1. S u p p o s e t h a t G = < a , b, c > i s a p - g r o u p ,

I ~ l = p ~,

Ib[,----p o, . ~ > ~ > t ,

l~[=p,

[~,b]=~,

[~, c] = [b, ~] ---- i. If in t h e c e n t e r ~(G) of the g r o u p G t h e r e e x i s t s a s u b g r o u p i s not a m a x i m a l c y c l i c s u b g r o u p of G, G / < r > m e t a c y c l i e , then [GI = 8 o r I GI = 16. If

]aPl

P r o o f . If t h e l e m m a i s f a l s e , t h e n p, t h e n n e c e s s a r i l y ~-~ . L e t xP = r a n d x = aCqbfile'gl. Then

(el. [3], p. 214). H e n c e , [bfll, a ~ I ] P ( P - ~ ) / 2 * 1, i . e . , p = 2 a n d l a l _<4, w h i c h i s i m p o s s i b l e . If aP = 1, t h e n IG] = 8. C o n s e q u e n t l y , laPl = p. L e t r ~ < x } , Ix[ = p2. But < e } i s a m a x i m a l c y c l i c s u b g r o u p of t h e g r o u p G. T h e r e f o r e w e m a y w r i t e G = ( such that G / < r> is metaeyclic. Then G is a group of one of the following types: i) G is a metaeyelic group; 2) G = <~-> X M, where M is a metaeyelie group;

3) G=(• 4) G=(• 5) G=(•

l a l = p m, l b l = p n, m ~ n ~ t , [ a l = 2 m,

m)3,

l a I = 2 "~, r a ~ 3 ,

[ a , b ] = T l = x a ~ W ~-1, e = 0 , 1 ;

lbl=2,

[a,b]~a-2"q, Tl=~a~2m-1, e = 0 , t;

b~=a ~-1,

[a, bl=a-2~i, Tl=Ta ~sm--1, e = 0 , t;

6) G l ~ ( X t i, aP~=b ~, s and t > 0, (T> i s a m a x i m a l c y c l i c s u b g r o u p of G a n d G ' = , k > 0. P r o o f . L e t G/<~-> = ( a ) ( b ) be a m e t a c y c l i c g r o u p . In v i e w of B 1 w e m a y w r i t e ~-~ B(G). If ~-~ ~(G), t h e n G i s a g r o u p of t y p e 2). T h u s we m a y f u r t h e r w r i t e ~- 6 r H e n c e d(G) = 2 and G = G / B ( G ) i s a g r o u p f r o m B 2. If <~-.> i s a m a x i m a l c y c l i c s u b g r o u p of G, t h e n G i s a g r o u p of t y p e 6), in v i e w of B 3. If <-r> i s not a m a x i m a l c y c l i c s u b g r o u p of G, t h e n i t s i m a g e <~> i s a l s o not a m a x i m a l c y c l i c s u b g r o u p of G. Since G / < ~ - ) i s a m e t a c y c l i c g r o u p , t h e n by L e m m a 1, p = 2 = m , n = 1 a n d by B2, ! G : G ' ] = 8. A s w a s noted in t h e p r o o f of L e m m a 1, < g > i s a m a x i m a l c y c l i c s u b g r o u p of G a n d t h e r e f o r e ~ ~ G'. T h e r e f o r e , [ G / <~-> : ( G / @ ' > ) ' l = 4 a n d a c c o r d i n g to [4], G=a/(v}=(a}(b},

[ a [ = 2 m', r a l ~ 2 ,

b - l a b = a -1+~'~1-~, e = 0 , 1, b ~ = l

o r b = a ~ 2 m l - 1 . But G =(X<~>) , b - t a b = a-lT1, r l = Ta~2m1-1 a n d e i t h e r b 2 = 1 , o r b 2 = a2m1-1 , b 2 = T, o r b 2 r a 2 m l - 1 . If b2 = 1 o r b2 = a 2 m l - 1 , then G i s a g r o u p of t y p e s 3) o r 4). I f b 2 = r o r b 2 = r a 2 m l - 1 , t h e n r e p l a c i n g b by b I = ba, w e o b t a i n a g r o u p of t y p e s 3) o r 4). The l e m m a i s p r o v e d . L E M M A 3. T h e p - g r o u p G i s a ( d , d ) - g r o u p if a n d o n l y if G / B ( G ) i s a ( d , d ) - g r o u p . P r o o f . T h e n e c e s s i t y in t h e l e m m a i s o b v i o u s . T h e s u f f i c i e n c y w i l l b e p r o v e d by i n d u c t i o n on the o r d e r of the g r o u p . L e t G b e a g r o u p of s m a l l e s t o r d e r s u c h t h a t G/B(G) i s a ( d , d ) - g r o u p , but G i s not a

608

( d , d ) - g r o u p . T h e n f o r r __- B(G), G <'r> i s a ( d , d ) - g r o u p s i n c e B ( G / < ' r >) = B(G) <~->/<~->. T h i s s h o w s that, t h e r e e x i s t s a m e t a c y c I i c s u b g r o u p ~ in t h e g r o u p G = G/ s u c h t h a t G/M i s an a b e l i a n g r o u p of r a n k d - 2. L e t H be the c o m p l e t e i n v e r s e i m a g e of the s u b g r o u p ~I. In v i e w of L e m m a 2 a n d by c h o i c e of the g r o u p G, H i s a g r o u p of one of the t y p e s 2)-6) of L e m m a 2 a n d G ' - - H. A s s u m e t h a t O ' ) i s not a m a x i m a l c y c l i c s u b g r o u p of G ' . A s i s o b v i o u s f r o m t h e s t r u c t u r e of g r o u p s of t y p e s 2)-6) of L e m m a 2, t h i s i s p o s s i b l e o n l y w h e n H i s of t y p e s 3) o r 5), w h e r e <~-> can be c o n t a i n e d o n l y in a c y c l i c s u b g r o u p ( h > of o r d e r 4andh# ~(H). S i n c e (T>___ @(H), t h e n d ( G ) = d . F r o m t h e f a c t t h a t G = H ( g 3 > . 9 .(gd) and of B(G) N Z(G) i s m a x i m a l cyclic in G'. Since H' is always cyclic and r - - @(G') -----B(G), then I H'I = p. T h e r e f o r e f o r t h e g r o u p H there are the following possibilities: 1) H = Q • 2) H = ( < a )

X
I[a,b][=p,

[al=p m,

I b l = p n, m _ > 2 ,

n_>l;

3) H = X <•>; 4)

H = ( < a ) X(-c,)) X ( b ) ,

[al = p m ,

lbl = p n , m_> n_> 1, [a, b] = r i.

It i s e a s y to e s t a b l i s h t h e f a c t t h a t G ' ___ ~(H) i t > . F r o m t h e m a x i m a l i t y of s i n c e G ' i s a b e l i a n . H e n c e G a ~ < 1 ) . H c a n n o t b e a g r o u p of type1),

s i n c e Z(Q) = ~ ( H ) <~ G. If H i s of t y p e 2 ) , t h e n G ' ~ _ ( a P pn-2 G, t h e n f o r n > 1 and t h e f a c t t h a t . If n = 1, t h e n G ' = r • @> -----Z(G), i . e . , G 3 = ( 1 ) r e m a i n i n g e a s e s f o r G 3 = ( 1 ) , f o r e x a m p l e , f o r t h e g r o u p H of

G' ~
(contradiction).

~1> ~

m-l,

bP n - t ,

r>.

S i n c e
=H

f o l l o w s t h a t a l s o <~ G and h e n c e ( c o n t r a d i c t i o n ) . W e a l s o show that i n the t y p e 4) f o r n > 1 w e o b t a i n


G' ~ a n d h e n c e

G3 =

<1 ).

The lemma is proved.

L E M M A 4 . S u p p o s e t h a t G = ( x l , . . . , x d ) i s a p - g r o u p a n d p d = i G:4}(G)I, d > 1. T h e n G ' =M'iM2' ([xl, x2] ) , w h e r e M 1 and M 2 a r e s u b g r o u p s of i n d e x p in G, w i t h M i = G ' < x t P , x2, . . . , X d ) , M 2 = G , ( x l , x2P, . . . , x d ) . Proof. Asiswell k n o w n [ 5 ] , G ' = (G3, [xi, x j ] j i , j = l , . . . , d } . l, et Mt and M2 be a s in t h e l e m m a . W e m u s t s h o w t h a t G 3 ~ MI'M2'. L e t f E G ' a n d g = f x ~ l . . Xd ~ d ~ G. If h E G ~, then [h, g ] = [ h ,

x~,...x~%][h, •

]][h, f, x , ~ . . . x ~ % ] = [ h ,

/], x ~ . . . x ~ ] [ h ,

x2~...x~%~][h, x,~,][h, x~,,x~...x~%J[h,

f], x~,] [[;~,

]]

/], x,~,, x~...a~,~,]

S i n c e e a c h f a c t o r b e l o n g s to M 1' o r to M2' , a n d h a n d g a r e a r b i t r a r y e l e m e n t s of G ' a n d G, r e s p e c t i v e l y , t h e n [G', G] = G 3 ~ MI'M2'. T h e l e m m a i s p r o v e d . L E M M A 5. S u p p o s e t h a t in t h e p - g r o u p G, r ~ ~(G), G / < v ) g r o u p of one of t h e f o l l o w i n g t y p e s :

-~ Vp~, Z(G) i s n o n e y c l i c .

Then G is a

1) G i s a n a b e l i a n g r o u p of t y p e (2, 1, 1, 1); 2) G ~ - ((a> X (oh>) • 3) G ~ ( ( ( a ) 4)

X
X (cr,>)X((y2))X(c~>,

G~DXV~,

[a]=p ~,

[a, c~z]~---i, [,7,, z~]---~

a',p>2;

p>2;

5) G ~ O • v..,; ~ [O, (o',}] 6) G ~ (0 X @,>) X <0".,,,',

--:----(~z

P r o o f . If G i s a n o n a b e l i a n g r o u p , t h e n in v i e w of r e s u l t !II, G = HC(H), w h e r e I HI = p3, H ~ = Q~ = ( T } . S i n c e Z(G) i s n o n c y c l i c , t h e n C(H) i s a b e l i a n a n d a s i m p l e c a l c u l a t i o n e s t a b l i s h e s the r e s u l t . T h e lemma is proved. P r o o f of the T h e o r e m . L e t G ~3f. proving the following lemmas.

F r o m the l e m m a 4 d i G ' ) _< 5. T h e p r o o f of the t h e o r e m r e q u i r e s

609

LEMMA

6. If G' E Jfand

Proof. Let G' ~Vps. dex p in the group G). Let

diG) : 2, then G'4=VpS;

In view of Lemma4,

in particular,

d(G') _< 4.

[Mi'[ = p2, Mi, N Mj' = (i),

i ~ j (M i a subgroup

of in-

G = , M~ = , M2 = , M s ' = ([c, d, c], [c, d, d, d], [c, d, d, d, c]>.

If [[c, d, d], d] - 1, then M 1' is cyclic (contradiction). d, d, c ] E M I'R M 2' = (1>. Consequently

Hence, M 1' = <[c, d, d]> X <[c, d, d, d]>. Hence [c,

[[c, d, d], c d ] = [ c , d, d, d][c, d, d, c][c, d, d, c, d] = [c, d, d, d] 6 M / [-1M/ = (t>, w h e r e M 3 = (contradiction).

The 1emma is proved.

LEMMA 7. If G EiTdand d(G) = 2, then G' cannot be an abeIian group of type (2, 1, 1, 1). Proof. L e t G = (c, d> and let G' be an abelian group of type (2, 1, 1, 1). It is n e c e s s a r y to a s s u m e that [[c, d]! = p2. As in the proof of L e m m a 6, it is e a s y to show that if M~ = G', M~ = G' and M/=([c,d,

c], [ c , d , c , c ] , [c,d, c,c,c]>.

In view of L e m m a 4, M i' N Mj' ~ ~(G') and without loss of g e n e r a l i t y we m a y w r i t e [ Ml'<[c, d] >[ = p4. Hence M,' = ( [c, d, d] > X ([c, d,d, d]><3G. It is e a s y to v e r i f y that [c ~, d] E [e, d]a[e, d, e]fiM l' f o r s o m e fi; in p a r t i c u l a r , [cP, d] E [c, d]P[c, d, c]YMt '. On the o t h e r hand, [c, d] E Ml'. Hence 1 a [c, d]P[c, d, c] T E Ml'. Note that by L e m m a 4, n e c e s s a r i l y [c, d, c] ~ M1' ([c, d] > and, in view of the m e t a c y c l i c i t y of M 1' and the choice of M1, [c, diP X [c, d, c] y r M l' {contradiction). T h e l e m m a is proved. LEMMA 8. Suppose that the group G E3g'k, d(G) = 2, p = 2. Then G' - V24. Proof. 6 and 7,

L e t G' ~ V24 and G = . If M 1 = ( G ' , c 2, d>, M 2 = (G r, 0, d2>, then just as in L e m m a s M ( = <[c, d, d]> X <[c, d, d, d]>and[c, d, c] ~ M / .

On the other hand, [c 2, d] = [c, d]2[c, d, c] E M 1' (contradiction).

The 1emma is proved.

LEMMA 9. Let G E3~, d(G) = 2. T h e n the following groups cannot be isomorphic to the e o m m u t a n t G' of the group G:

1) D X 0 9 , p > 2 ; 2) D X Vr p > 2 ; 3) (( X <(1~>) )', ) X (a~>,

p > 2;

]a] =p2,

[a, zd

= i,

4) (((a> X @,>) X Vv ~, p > 2, Ia] =p2, [a, a,] -~ aP;

5) (( x

<~>) x

<:~>, lal-- 4, b, d = ~ ;

6) (((a> X <~>) h <•,>) X (a.~>, [ a [ = 4 , [a,, a ] = ~ ; 7) ((X(~>) ~, ) X , l a l = I b l = 4, b, b] = ~; 8) (Q X <'0) IX <~>, [Q, <~>] = .;

m ((<,~>x<,~,>) x<-O) x<,~2>,

610

I~1 = ~ ,

[~,~,1 =a s, [~,~d=~,

[~,,

z21 = a~;

P r o o f . If G ' is a g r o u p of type 1), then [ Mi'l = p2. M o r e o v e r , [c, d] ~ Z(G'), o t h e r w i s e G' -----Z(G') c G' (contradiction). Since G' = ([c, d], G 3) [5], then G 3 ~ Z(G') ([c, d ] ) . Let ( [ c , d], [c, d, d]) ~- D. Since [c, d , d ] ( M ' , wherelVi = G ' ( c p, d ) , then ( [ c , d , d ] ) • D' = M ' < ~ G. H e n c e [ G , ( [ c , d , d ] ) ] ~_D'=G". On the o t h e r hand, in the g r o u p G the equality [c, d, d] cd = [c, d, d] d c [ c , d ] holds. Hence [c, d, d] [c, d, d, c] [c, d, d, d] [c, d, d, d, c] = [c, d, d]t~. q[c, d, d, d] [c, d, d, c] [c, d, d, d, c] and [c, d, d][d, c] = [c, d, d] (contradiction). If G' is a group of type 2), then from diction).

T h e r e f o r e G' cannot be a g r o u p of type 1). Lemma

4 it follows that M' = D' X (or} and d(G') _ 3 (contra-

Let G' be a group of types 3) or 4). Then on the one hand, ~(G') = B(G). On the other hand, (aP} ~M', where M is an arbitrary subgroup of index p in the group G. By Lemma 4 and from the fact that G (Y~, it follows that M' is an abelian group of type (2, i), since Z(M') is a noncyclic group, if M" ~ (1) [6]. Moreover,

M" ~ N < 3 G , I N l = p ~, N ~ G ' . Hence, by B3, N = B(G), but exp (B(G)) = p, and exp (N) = p2. This e o n t r a d i c t i c n shows that G ~ cannot be a g r o u p of t y p e s 3) o r 4). If G' is a g r o u p of t y p e s 5) o r 6), then as in c a s e 1), we m a y show that ([c, d], [c, d, d] ) -~ H, w h e r e H is a g r o u p of type 5) of L e m m a 9, and G = ( c , d}. On the o t h e r hand, f r o m the equality

[c, d, d]cd = [c, d, d]~0~o, ~. As in c a s e 1), we obtain that [[c, d], [c, d, d]] = 1 (contradiction) and G' cannot be a g r o u p of t y p e s 5) or 6). If G' is a g r o u p of type 7), then it is e a s y to show that in such g r o u p s t h e r e a r e e x a c t l y t h r e e abelian s u b g r o u p s of type (2, 1, 1, 1) and one of t h e s e c o i n c i d e s with B(G) by B 3. Then G ' / # B ( G ) is a g r o u p of type 6) (contradiction). L e t G' be a g r o u p of type 8) and Q = ( a ) ( b ) . If a ~ = aT, b cr = b, then H t = (a}(b} • (@ and H 2 = ( b } X (~-) • (a} a r e c h a r a c t e r i s t i c s u b g r o u p s in G' and in v i e w of B), H~ = B(G) = H 2 (contradiction). Hence G' cannot be a g r o u p of type 8). F i n a l l y , let G' be a g r o u p of type 9), w h i c h has the fello~4ng s u b g r o u p s of index 2:

//, -----((a) x @,)) • ('0, H~ ----((a~,~) x @ # ) • @ %

.~,---- (: x )x<-r>,/& =(X<'O)x , H~ = (<~cr~>•

x X<'O) x <,:r~>,

H, = X<'O X (~,> X (o'~>.

Since H I is isomorphic to only the subgroups H 2 and H3, then one of these subgroups is invariant in G. Besides, H 7 is a characteristic subgroup of G'. Therefore (cf. B3) H I = B(G) = H 7 (contradiction). The Lemma is proved. The proof below of the theorem proceeds in several steps. Y~, d(G) = 2, not satisfying the conclusion of the theorem.

Let G be a group of minimal

order from

i) B(G') = (I). Infaet, if (:r)~ B(G'), thenG/Z~}~J~and IG/(~')I < [G]. Hence by the induction hypothesis, G'/(T} is a group of one of the three types of the theorem. In view of Lernmas 3) and 5)-9) we obtain a contradiction to the choice of the group G. 2) G" ~- Vp2. In fact, G" ~_ M', where M is an arbitrary subgroup of index p in G. By hypothesis, is a metacyclic group and, in particular, G" is metacyelic. By I), G" is an elementary abelian group. H e n c e IG"I -< pZ and we obtain two c a s e s : G" = ( 1 } and ]G"[ = p. If G" = ( ! ) ,

M'

then in view of L e m m a s 6-8 we obtain a c o n t r a d i c t i o n to the choice of the g r o u p G.

L e t [G"] = p. T h e n G ' / G " is an abelian g r o u p of t y p e s 1) and 2) of the t h e o r e m , s i n c e the c a s e G ' / G " -~ Vp4 is i m p o s s i b l e , in view of L e m m a s 5 and 9. tf G ' / G " is a m e t a c y c l i c group, then by the choice of the g r o u p G and by [7], G' is a g r o u p of the f o r m ((a>X@->)X, [al~---p', ] b l = p % m ~ n : ~ l ,

[a~ b ] = T

w h i c h is a n o n m e t a c y e l i c g r o u p of M i l l e r - M o r e n o . 61t

If p > 2, t h e n P(G') = ( a P , bP><:] G a n d G ' / P ( G ' ) ~- D, which i s i m p o s s i b l e icf. [6]). If p = 2, t h e n G ' /@B(G) i s a g r o u p of type 5) of L e m m a 9, a n d i n v i e w of t h i s l e m m a we o b t a i n a c o n t r a d i c t i o n . T h e r e f o r e d ( G ' / G " ) = 3 a n d d(G') = 3. U s i n g r e s u l t IlI, we o b t a i n that G' = H ( z ) , w h e r e z E Z(G'), a n d H is a n o n a b e l i a n g r o u p of M i l l e r - M o r e n o . By the c h o i c e of G, H i s a n o n m e t a c y c l i c group. L e t ~ E K 1 = . F o r p > 2, K 1 = P I G ' ) < 3 G a n d G ' / K l is a g r o u p of type l) of L e m m a 9 , w h i c h i s i m p o s s i b l e . I f p = 2, t h e n ~-~ K 2 = ( a 4, b 4, z2> = a~(Z(G')) a n d G ' / K 2 i s a g r o u p of t y p e s 6) o r 7) of L e m m a 9 a n d i n v i e w of t h i s L e m m a 9 a n d i n v i e w of the l e m m a we o b t a i n a c o n t r a d i c t i o n . C o n s e q u e n t l y , ~ = a~Op~bfiopfizYOp y, ~ , / 3 , y > 0, a0, fi0, Y0 ~ 0(p).

L e t x = m i n ( a , fi, y).

If

(aaoPCL--Xb~~

VoP'~-x)P~= T [b~", aa'] P~+~-xP(~-i)/~)

= ~" ([3], p. 214), then, f o r e x a m p l e , f o r x = a we o b t a i n

G' = ((aaob~oP~-XzYopV-X}h (b>) (z) a n d G' i s a ( 3 , 3 ) - g r o u p , w h i c h c o n t r a d i c t s t h e c h o i c e of t h e g r o u p G. H e n c e p ~-- 2, x = t -~- a -~ ~, G' ~--- (( X <~>) s (~})(z>, where

T h e n K = # ( G ' ) Z ( G v) ~ ( z ) , t h e n G ' / ~ ( K ) i s a g r o u p of type 6) of L e m m a 9, which i s i m p o s s i b l e by this l e m m a . S i n c e ( a 2) ~ ( z ) , i . e . , a 2 = zt02t, t o ~ 0i2), t > 1 a n d t h e n ((a-lzt02t-l> X (~-> h (a> ~- D a n d G ~ is a ( 3 , 3 ) - g r o u p , w h i c h c o n t r a d i c t s the c h o i c e of the g r o u p G. T h i s c o n t r a d i c t i o n shows that the c a s e [G"I = p i s i m p o s s i b l e , a n d t h i s p r o v e s the a s s e r t i o n of our T h e o r e m 2). 3) G T is a g r o u p of one of the f o l l o w i n g t y p e s :

( x x <~>)<~>,

I~l

= p~, Ibl ---- P",

((<~>X)X<'O)<~>, l ~ l - - p ~,

(((a) X ( a ) ) X(b))(c),

I b l = p o, , ~ > 2 ;

l a l = p "~, I b [ = p ~, m ~ ;

[a, b] = cr, [(r, b] = 1, a = ~ - a e l p m - l b s 2 p n - 1 , ~I, s2 r e s i d u e s m o d p , H = ( a , b, ~-> the c o m p l e t e i n v e r s e i m a g e of the i n v a r i a n t m e t a c y c l i c s u b g r o u p M of the f a c t o r - g r o u p

G' = G'/(.~} = M(8), (-~) ~ ~ ( G ' ) , d ( V ' ) = 3, ( e ) ( x ) / ( ~ > = (~). I n faet, by c h o i c e of the g r o u p G, i n view of L e m m a s 2, 5, 8, 9 a n d a s s e r t i o n 2), G ' / ('c> i s a (3,3)g r o u p for <~>E @(G'). In M = < a >< b > , t h e n the c o m p l e t e i n v e r s e i m a g e w i l l be as i n the g i v e n t h r e e c a s e s of a s s e r t i o n 3) or = Q X < ~ >. If we have the l a t t e r e q u a l i t y , t h e n Hi = (Q i s i m p o s s i b l e [6]. A c c o r d i n g to HI a n d a s s e r t i o n G it follows f r o m t h i s that Hi = < 1 ) a n d K = Z(Q) c e n t e r of the g r o u p G'/. H e n c e
4).

>~<~->) N , o t h e r w i s e Z I G ' / < T > ) i s c y c l i c , w h i c h 2) we m a y w r i t e [G', < c>] = <~>. By choice of the g r o u p X < ~ > • < c > ~ G as the c o m p l e t e i n v e r s e i m a g e of the G ' / < c2> a r e g r o u p s of type 8) of L e m m a 9 ( c o n t r a d i c t i o n ) .

4) L e t G " =<~- >X <~r>. T h e n t h e r e e x i s t s i n G" a n e l e m e n t v ~ 1, s u c h t h a t a E H , T h i s f o l l o w s f r o m L e m m a 1, if a a n d b a r e e x c h a n g e d i n H.

v E , v

5) Suppose that t h e r e e x i s t s i n G" a n e l e m e n t w s u c h that G" -- X and. v i s a n e l e m e n t a s i n T h e n one of t h e f o l l o w i n g a s s e r t i o n s i s valid: t h e r e e x i s t s a n e l e m e n t h i n G ' s u c h t h a t h ~ ~(G'), ( w > c < h>, a n d m o r e o v e r , e a c h s u c h e l e m e n t h

has the f o r m h = a~ObfioPflcYoPTrS, w h e r e a0, /3o, Y0 ~ 0(p), /3 a n d y ~ 0. (co} i s a m a x i m a l c y c l i c s u b g r o u p of G' for each w for which G" = X (co}. A s s u m e that i n the g r o u p G ' t h e r e e x i s t s a n e l e m e n t g s u c h that gP = co, w h e r e co ~ G" \ < v > . G" ~-- Z(G'), we m a y w r i t e g = ak~176176

612

Since

ko, r0, l0 ~ 0(p), k > 0 a n d f o r x = r a i n (k, r , l) a n d for the

element f =

akopk-Xbropr-XcloPl-X

i t is e a s y to v e r i f y that w = fpX+t a n d f ~ ~(G"). If x = r , then we c o n -

s i d e r the the s u b g r o u p H 0 = ((a) • (co))(f).

Since G ' = H o ( c ) , t h e n by the c h o i c e of G, H0 carmot be a m e t a -

c y c l i c g r o u p ; i n p a r t i c u l a r , [a, f] = vco. L e t Ial = pro, m > I a n d m -< x + 2. T h e n we c o n s i d e r the e l e m e n t d = alP x + 2 - m Since

(~m-~)/~

d~-~ = a ~'~-~/~+~ [/, a ]~+~ ~'~-~

t h e n i n c a s e s = pX+2 - - m p m - - l ( p m - - i _ 1 ) / 2 ~ O(p) we o b t a i n that It 0 = ( a ) tion). H e n c e s ~ O(p), which i s p o s s i b l e only i n c a s e p = 2 = m a n d x = O.

X ( a ) is m e t a c y c l i c (contradic-

Thus,

= af a n d G' = H o ( c ) .

Two c a s e s a r i s e .

A 1. ~" ~ {vco}. T h e n G ' G ' / ( T ) - ~ D { ~ ) , w h e r e z E Z(G'). W e m a y w r i t e a 2 = T a n d i T ) ( c ) i s t h e c o m p l e t e i n v e r s e i m a g e of the s u b g r o u p ( ~ ) . Since Z(G') i s a n o n c y c l i c gToup, t h e n H 0 rl ( c ) ~ ( "r ) a n d b y t h e c h o i c e of the g r o u p G, H 0 rl ( c ) = (1}. C o n s e q u e n t l y , ( % ~-c0, c ) is i n v a r i a n t i n G, a s the c o m p l e t e i n v e r s e i m a g e of Z(G*). T h e r e f o r e ( c 2 ) 2. We c o n s i d e r the f a c t o r g r o u p G' = G , / ( c 4 ) If ~2 = 1, then i t i s e a s y to v e r i f y t h a t the f o l l o w i n g s u b g r o u p s have i n d e x 2:

a r e c h a r a c t e r i s t i c i n G ' a n d i n view of Be, MI = B(G) = I~2, w h i c h i s a c o n t r a d i c t i o n . w r i t e ~2 = 0. In this way, a s a t s o f o r a z = 1, we o b t a i n a c o n t r a d i c t i o n . A 2. ~- = vw. c a s e s . Secondly, t r a r y c a s e for a n a c o n t r a d i c t i o n to

T h i s a l l o w s us to

F i r s t of all, i n t h i s c a s e we m u s t show that a z ~ Z(G), o t h e r w i s e we o b t a i n p r e v i o u s the s u b g r o u p ( ' r ) c a n n o t b e a m a x i m a l c y c l i c s u b g r o u p i n the g r o u p G', s i n c e the c o n a r b i t r a r y s u b g r o u p M of i n d e x 2 i n the g r o u p G, M' ~ G s. U s i n g L e r n m a 4, we o b t a i n the f a c t that d(G') = 3.

Set gl E G% g 2 = T, gl = a~o~~ c.~o2~ ~v, %, ~0 ~f~ 0 (2), y = 0, i ; w h e r e ~ , /3 a r e n o n n e g a t i v e i n t e g e r s . If 7 = t a n d 2fi~0 -= 0(2), t h e n G' = ( ( g l ) s w h i c h c o n t r a d i c t s the c h o i c e of the g r o u p G.

is a (3,3)-group,

If i~ = I a n d 2tiff 0 =-1(2), w h i c h by the c h o i c e of the g r o u p G n e c e s s a r i l y m u s t s a t i s f y the e q u a l i t i e s [a, gl] = a 2 r a n d [ a g , gI] = a2 a n d h e n c e [g, g] =~-. In this c a s e

~' = ( (x X <~> i s a g r o u p of type 9) of L e m m a 9 ( c o n t r a d i c t i o n ) . T h e r e f o r e , I/ = 0. Set g = m i n (a,

~),

g~ ~ a~o2~-~j b~o# -~'.

If y = p, t h e n s i n c e the s u b g r o u p s (a, T, g2 > a n d < a g , ~, g2) by c h o i c e of the g r o u p G c a n n o t b e m e t a c y c l i c , we o b t a i n [a, g2] = a2% [ g2[ = 4, a n d t a g , g2] = a2. H e n c e [g, g2] = T, i . e . , G' i s a g r o u p of type 9) of L e m m a 9, w h i c h i s i m p o s s i b l e . T h e r e f o r e y ~ fi, i . e . , y < fl a n d y = a . S i n c e ~a 2 =

61ZC~~

~

T.

t h e n H ol~ ( c ) = (a27>. S i n c e ( c 4 ><3G, G " [ ~ Z(O), t h e n e 2 =a2% i . e . , trio 2 f i + l - y =a2 r : e2" T h e r e f o r e fi - y + 1 = 1, i . e . , y = fi ( c o n t r a d i c t i o n ) . A s s e r t i o n 5) i s p r o v e d , s i n c e the c a s e x = l i s a n a l o g o u s .

613

Since in 4) v ~ (T }, then from 6) If ( 7} is a maximal Lemma

5) follows:

cyclic subgroup

In the contrary case, M' _ G3, where 4, d(G') < 3 (contradiction).

in G', then G" ~-- Z(G'). M is an arbitrary subgroup

of index 2 in the group G.

7) If G" =(v} X (w}, v E (a}, v ~ (T}, then in G" there exists an element and (T O } is a nonmaximal cyclic subgroup of G'.

Then by

T O such that G" = (v} X (T0)

If (T} is a nonmaximal cyclic subgroup, then the assertion is proved. Let (T} be a maximal subgroup of G'. By assertion 6), G" ~-- Z(G). We consider the factor-group G'/(v). In this group let (7}(v> / iv)/(v) is nonmaximal in G,. Let (~} ~(g), (g>= p2. Consider the subgroup ((a} X (T))(g}, where(v}(g} is the inverse image of the subgroup (g}. It is easy to verify that gP = TelVe2 and e I ~ 0(p). In that case gP will also be the desired element T O . 8) If G" = (v) X (w}, v E (a), then there exists an element T0 as in assertion the first assertion of 5) is valid. torr~

9) If v is an element as in 8), then there exists an element 0(p).

7) and therefore for Y0

g of G' such that v~ (g), g = aSbtc r and

A s s u m e t h a t e a c h e l e m e n t g of G ' f o r w h i c h ( v } _ (g), h a s t h e f o r m g = aSbPt~cPYl. F u r t h e r , if M i i s a s u b g r o u p of i n d e x 2 in G, t h e n G" = ( v } • (~'0} --~ M i ' a n d M i ' i s a m e t a c y c l i e group: M i ' = (ci) ~ (di> a n d [Mi"[ _ a n d M i ' = ((ai> • (Ti})(bib. If M i ' -~ T, t h e n in T a l l c y c l i c s u b g r o u p s of o r d e r two a r e c h a r a c t e r i s t i c , a n d T / ( b l 2} -~ D ( c o n t r a d i c t i o n ; [6]). In t h i s c a s e a i = a S i b P t i c P 7 i a n d in v i e w of t h e h y p o t h e s i s , b i = a k i b P l i e P r i , k i ~ 0(p) a c c o r d i n g to a s s e r t i o n 8). F r o m L e m m a 4, G ' = MI'M 2' ( [ e , d]}, if ' ( c , d } = G. T h e r e f o r e , d(G') _< 2 ( c o n t r a d i c t i o n ) . T h i s p r o v e s a s s e r t i o n 9). W e c o n s i d e r t h e s u b g r o u p H = ((h> • (v)) ( g ) , w h e r e ( 7 0 } c ( h } a n d T0 a n d h a r e e l e m e n t s a s in a s s e r t i o n 8), a n d t h e e l e m e n t g i s a s in 9). L e t g = aSbc r . S i n c e t h e c o n g r u e n c e ~0 x ~ - s ( m o d p ) i s s o l v a b l e f o r x, t h e n ~ H and G' : [I . In t h i s c a s e , a s in t h e p r o o f of a s s e r t i o n 5), w e o b t a i n a c o n t r a d i c t i o n e i t h e r to t h e c h o i c e of t h e g r o u p G, o r , b y L e m m a 9, to the u n i q u e n e s s of B(G) in G ' in T h e o r e m B 3. T h e t h e o r e m i s p r o v e d . T h e a u t h o r e x p r e s s e s h i s d e e p g r a t i t u d e to A. I. S t a r o s t i n f o r p o s i n g t h e p r o b l e m , and f o r his h e l p f u l guidance.

LITERATURE i~

2. 3. 4. 5. 6. 7.

614

CITED

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