= « 0
= *>0
— t>oa0b0 a0 bo&Q-- i - i
A m o d erately difficult coset en u m eratio n by co m p u ter shows th a t th e preim age of Ho w ith a p resen tatio n on ao an d bo w ith relations
( 4 - 3 1 ) 1 = (i>»a0 2)2,
a0 bo a o bQ ao bo ao ao
has o rd er 1296, an d th u s is a deficiency zero presen tatio n for Hq-
T h e subgroup of Hq generated by a\ = a\ and bi = aob^1 has index 2, and is th u s th e derived group of Ho- A two generator, two relation presen tatio n for H i = (a i 5^i) is o b tain ed by using a m odified form of coset enum eration ( see B eetham and C am pbell (1976)):
(4 • 32) H i = { a i . b ß a l = ((ai&i)3)61, = ( 6 r 3 r 1 6 ^ ) ,
and so we also have a deficiency zero p resen tatio n for a finite group of soluble length five. This is th e group G3 defined in th e previous section.
C ontinuing down th e derived series of Ho, H[ = ( a i, 63, b i a i b ^ 1) is of order 216 an d has soluble length four. Using co m p u ter im plem entations of th e R eidem eister-Schreier alg o rith m and Tietze tran sfo rm atio n s, a deficiency zero p resen tatio n for H[ of order 216 is obtained on generators a-i — a\ and b 2 — b ^ l a \b \ w ith relations
1 C L -y b * ) C L ^ )b -y CL*y b o
--- C L-ybty C L-ybty CL-y b ^ C L o b - y .
T h e following th e o re m p rovides a n o th e r ex am p le of a fin ite g ro u p of d e ficiency zero havin g soluble le n g th six. T h is g ro u p h as a so m ew h at different s tr u c tu re from th o se given earlier in th is section.
T h e o r e m 4. Let G be generated by x and y w ith the following defining rela tions:
(4 ■ 34) 1 4 —4
1 = x y
= x 2y x y x y ~ 1x ~ 1y ~ 1x ~ 2y.
G is a g ro u p of o rd e r 312000 hav in g soluble le n g th six.
P r o o f:
C oset e n u m e ra tio n over th e id e n tity show s th a t G h as o rd e r 312000. We d e te rm in e th e s tr u c tu re of G below . T h e d eriv ed g ro u p G' of G is g e n e ra te d by y x , x y, a n d y ~ l x y 2 a n d has in d ex eig ht in G. W e will show th a t G' is iso m orphic to G2, of o rd e r 39000, defined in th e p rev io u s section.
A set of Schreier g e n e ra to rs for G' is
y x , b = x y, a x , d = bx~ \ cx , f = ex , f \ h = d x~ \ a = c = e = 9 = i =
a n d using th e R eid em eister-S ch reie r a lg o rith m , a p re s e n ta tio n for G' is
1 = bghd = c e f g = df gh
x l 2 7 - 2 2 - 2 - 1
(4 • 35)
= ab2 d ~ 2 = a c e f i ~ l = acei ~1b = aci~^ db = a i ~l hdb = bd2h~2 = c2e~2r 1 = e 2r 2i - 2g ~ 1=
dh2 g ~ 1i ~ 1 g ~ 1i ~ 1 = f 2( i - l g - l )2i - l h - ' iThis presen tatio n contains m any re d u n d a n t generators and relations. We now simplify th e presentation. Using th e first relation to elim inate h = g ~ 1b~1 d ~ 1 and simplifying th e p resen tatio n , we ob tain
1 =
b f - 1
= a g ~l i~ 1 = a i ~l g ~ l = b f ~ 1e ~ 1d = cef g = d i e ~xi ~ l = a2b~2c(4-36)
= a 2c - 2e ~1 = a b 2d ~2 = acei ~1b = b2i f ~ 1f~ 1 2 —2 r —1 • r 2 —2 = c e f = a i f e = agd~2b~l d ~ 1 i = bd2(dbg)2Next we elim inate / and simplify th e presen tatio n to o b tain
1 = 2 = d e-1 = a g ~1i~ 1 = a i ~ l g ~ l = bgce = di e ~l i ~ l
(4-37)
= a2fc-2c = a2c - 2e - i
= ab2 d ~ 2 = acei ~l b — be2 c~2 = aib2e~1e~ 1 = agd~2b~l d ~ 1 i = bd2 dbg dbgObserve th a t we have now o b tain ed a proof th a t x 8 = 1 holds in G. Elim inate i, e and g to ob tain
(4 -3 8 ) 1 = acdb _ „ 2 l—2 = a o c = a 2c~ 2d~ 1 = ab2d ~ 2 = bd2c~ 2
Using th e first relation to elim inate d
(4 -3 9 )
1 = a2b~2c = a2c ~ 2bac = ab3 acbac = acbacb- 1 c2b
From the first relation, ca2 = 62. S u b stitu tin g this into the second rela tion, we obtain ab2c~ 2b = 1. From th e th ird relation, we have acbac = b~z a. S ubstitu tin g this into th e fo u rth relation, we also o b tain ab2c~ 2b = 1, and so we ob tain a deficiency zero p resen tatio n on th ree generators.
(4 • 40)
1 = a~ b 2c
= ab2c~ 2b
= ab2bacbac
Finally, elim inate c to o b tain
(4 -4 1 ) 1 = a2b~1ab2a2b~2