Groups with Cyclic Factor Groups

T h is section is m o tiv a te d by th e fact th a t each polycyclic g ro u p h as a n o rm al s u b g ro u p such th a t th e c o rresp o n d in g q u o tie n t g ro u p is cyclic. T h e in te n tio n of

th is sectio n is to derive co ndition s u n d e r w hich a g ro u p G can b e b u ilt fro m a g ro u p N a n d a cyclic g roup C such th a t G h as a n o rm a l s u b g ro u p iso m o rp h ic to N a n d th e q u o tie n t g roup is isom orp h ic to C . L a te r, th is will be u sed in o rd er to te s t a given polycyclic p re s e n ta tio n for consistency. F or different acc o u n ts of th e th e o ry in th is section see Z assenhau s (1958, I I I .7), S co tt (1964, 9.7) a n d Sims (1993).

In th e follow ing let C ^ be th e in fin ite cyclic g ro u p a n d le t z b e a g e n e ra to r

O f Coo.

3.2.1 Lemma Let G be a g ro u p w ith a n o rm a l su b g ro u p N such th a t G / N

is cyclic. T h e n G is a h o m o m o rp h ic im ag e of a sem id irect p ro d u c t CooCKiV. P roof: Let x E G such th a t G = ( x , N ) a n d let a b e th e a u to m o rp h is m of

N in d u c e d by x v ia co n ju g atio n . D efine a h o m o m o rp h ism p fro m Coo in to th e a u to m o rp h is m g ro u p of N by m a p p in g z to a . Now th e sem id irect p ro d u c t

H = CootX iV w ith resp ect to th is actio n of C<*, on N can be fo rm ed . T h e m ap

0 : H — >G ( z l, ' l l ) I--- > x lu

is a h o m o m o rp h ism , w hich is obviously su rjectiv e, as th e follow ing c a lc u la tio n for

i , j E Z a n d U i , u 2E N shows.

( ( z l , - u1) ( z J , ' u2) ) V> =

3

2

T h e elem ent ( z l , u) of H is m a p p e d by -0 to th e id e n tity in G if a n d only if x l — u ~ l . If som e p o sitiv e pow er of x lies in N , let k be th e sm allest p o sitiv e in te g e r such th a t x k E N an d let u = x k . T h e n th e kernel of 0 is g e n e ra te d by ( z fc,i4_ 1 ). T h e g ro u p G is a sem id irect p ro d u c t of (x) a n d N if a n d only if

u = 1. If x l ^ TV for all i £ Z, th e n t/j is an iso m o rp h ism a n d G is iso m o rp h ic to a sem id irect p ro d u c t of (x) a n d TV.

Let TV be a g ro u p an d let z e-> a be a h o m o m o rp h ism fro m in to th e a u to m o rp h is m g ro u p of TV. F u rth e rm o re , let H b e th e sem id irect p ro d u c t of C Q0 a n d TV w ith resp ect to th is h o m o m o rp h ism . If G is a g ro u p w ith TV as a n o rm al su b g ro u p such th a t G / N is cyclic a n d if G h as an elem en t x g e n e ra tin g a su p ­ p le m en t to TV a n d actin g on TV v ia co n ju g a tio n as a , th e n G is a h o m o m o rp h ic im ag e of H. T h e k ernel of any such h o m o m o rp h ism h as b een d e te rm in e d in th e p rev io u s p a ra g ra p h . T h erefo re, in o rd e r to d e te rm in e all such g ro u p s (V, it is suf­ ficient to d e te rm in e all possible n o rm al su b g ro u p s of H th a t are g e n e ra te d by an elem ent of th e fo rm ( z k , u ~ l ) w here k is a p o sitiv e in te g e r an d u an elem en t of TV.

Let Z be th e cyclic g ro up g e n e ra te d by ( z k , u ~ 1). T h e n Z is n o rm a l in H if a n d only if ( z l , v ) - 1 ( z fc, ) ( z l , v) £ Z for all i £ Z a n d v £ TV. C o n sid er

{ z \ v ) ~ \ z k , u ~ l ) ( z \ v ) = ( z ~ l , u _1a _ t ) ( z /c,'a _ 1 ) ( z 1,'u) = ( z ~ l , u - 1Q " t ) ( z l+ \ ( ^ - 1a t )u) = ( z fc, ( u _1 a l )u).

B ecause z has in fin ite o rd er, th e la st te rm in th is ch ain of eq u alities is an elem ent of Z if a n d only if

(v~* a k ) ( u ~ l a l )v = u ~ l for i £ Z a n d v £ TV. (*)

S e ttin g v to th e id e n tity of H a n d i to 1 show s th a t u a = u. F ro m th is a n d specializin g i to 0 in th e eq u a tio n above it follows th a t v a k = u ~ 1v u for all

v £ TV.

O n th e o th e r h a n d , if u a = u a n d v a k = u ~ 1v u for all v £ TV, th e n

v a k = [ u ~ l a l ) vu for all v £ TV a n d all i £ Z. T h is show s th a t th e se tw o co n d itio n s are equivalent to (*).

3.2.2 Lemma T h e su b g ro u p of th e sem id irect p ro d u c t Coo X TV g e n e ra te d by ( z fc,'u ~ 1) is n o rm a l if a n d only if th e co n d itio n s

u a = u

v a k = u ~ 1v u for all v £ TV

are m e t.

If [ z k , u ~ 1) g en era tes a n o rm a l su b g ro u p Z7, th e fa c to r g ro u p (C o o X TV)/Z7 h as a n o rm a l su b g ro u p of in d e x k iso m o rp h ic to TV.

Let TV be given by a finite p re s e n ta tio n (X ,7 £ ). For each elem en t x £ X

th e re is a w ord w x in X such th a t x a = w x as elem en ts of TV. T h e follow ing is a p re s e n ta tio n for H :

^ X U {z}, 7ZU { z l x z = w x I x £ X .

Let k be a p o sitiv e in teg er an d u a w ord in X such th a t u a = u as elem en ts of TV a n d v a k = u ~ l v u as elem ents of TV for all w ords v in X , th e n

^ X U { z } , 71 U { z -1 x z = w x I x £ X } U { z fc =

is a fin ite p re s e n ta tio n for H / {{zk, u~ l )). C o m p a re J o h n so n (1990, C h a p te r 10.2.). As a first a p p lic a tio n of th e above th e o ry we prove th e follow ing th e o re m .

3.2.3 Theorem Let (A , S ) be a polycyclic p re s e n ta tio n in w hich all c o n ju g ate re la tio n s are triv ia l. T h e n (A, 5 ) is co n siste n t.

P roo f: Let G b e th e g ro u p defined by (A,«S) a n d let A' = A \{ a j} a n d S ' b e th e set of re la tio n s of S th a t do n o t involve a i .

A ssum e th a t (A ', S ' ) is co n siste n t. Let a j act triv ia lly on th e g ro u p G 1

defined by (A ', S ' ) . If a : has no pow er re la tio n in S th e n (A , S ) is a co n sisten t polycyclic p re s e n ta tio n by th e p rev io u s le m m a b eca u se G / G ' = Cqq. If a j h as a pow er re la tio n th e n th e two co n d itio n s of th e p rev io u s le m m a are triv ia lly satisfied a n d (A , S ) is a co n sisten t polycyclic p re s e n ta tio n b eca u se G / G ' = C k• Now th e