Computing presentations for finite soluble groups

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C o m p u tin g P r e s e n t a t io n s for

F in it e S o lu b le G r o u p s

A lic e C . N ie m e y e r

July 1993

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D e c la r a tio n

T h e w o rk in th is th e s is is m y ow n u n less o th e rw is e s ta t e d .

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A c k n o w le d g m e n ts

I t is w ith g re a t p le asu re th a t

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ta k e this o p p o rtu n ity to th a n k m y supervisor, D r M .F . N ew m an , for his inv alu ab le guidan ce a n d sup erv isio n d u rin g the course of m y stu d ie s.

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am very g ra te fu l for all he has ta u g h t m e d u rin g m a n y hours of d iscu ssio n a n d for his co n sid ered a n d careful advice an d e n c o u ra g e m e n t.

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th a n k m y ad visors D r L.G . K ovacs an d D r E.A . O ’B rien for m any very h elp fu l discussions a n d g en ero u s assistance.

D u rin g th e p a s t th re e years D r E .A . O ’B rien has offered m e a lo t of enco ur a g e m e n t a n d s u p p o rt w hich has been highly valuable to m e a n d for w hich I express m y sincere g ra titu d e .

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am deeply in d e b te d to D r M .F . N ew m an an d D r E.A . O 'B rie n for sharing w ith m e th e ir ex perien ce in w ritin g a n d for teach in g m e so m u ch by th e ir u n tirin g efforts a n d c o n stru c tiv e su gg estio n s.

W ern e r Nickel has alw ays resp o n d e d w ith g re a t e n th u sia sm in m a th e m a tic a l discussio ns to g e th e r.

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th a n k h im for his co n tin u o u s in te re st in m y w ork and the m a n y discussio ns we h a d a b o u t it.

W ith g r a titu d e

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acknow ledge th e help

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received from W e rn e r Nickel and M ichael S m ith in p ro o f re a d in g th is thesis.

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am also g rate fu l to M ichael S m ith for his te ch n ica l advice.

D r C .R . L e e d h a m -G re e n offered som e very im p o rta n t su g g estio n s and I th a n k h im b o th for th e se su g g estio n s a n d for som e very in spirin g discu ssio n s.

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I t has been a p leasu re to b e able to use D r S. L in to n ’s im p le m e n ta tio n of a v ecto r e n u m e ra to r. I th a n k h im for m a k in g it available to m e a n d for his excellent m a in te n a n c e of th e code.

T h e advice a n d help I received fro m P h ilip p a H arv ey a n d A n n e tte H ughes in m a n y of th e a d m in is tra tiv e ta sk s d u rin g th e tim e of m y P h .D . co u rse is very m u ch a p p re c ia te d .

I th a n k m y fam ily for th e ir love a n d s u p p o rt.

W ith deep g r a titu d e I th a n k m y frien d s, in p a r tic u la r M e h m e t K a ra n , for th e ir help a n d s u p p o rt in th e n o t alw ays easy tim e of th e la st 12 m o n th s.

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A b s t r a c t

T h e w ork in th is thesis w as carried o u t in th e a re a of c o m p u ta tio n a l g ro u p theory. T h e la tte r is concerned w ith designing alg o rith m s a n d developing th e ir p ra c tic a l im p le m e n ta tio n s for in v estig atin g p roblem s reg ard in g g ro u p s. A n im p o r ta n t class of g ro u p s are finite soluble groups. T hese can b e d escrib e d in a c o m p u ta tio n a lly convenient w ay by pow er co n ju g ate p re s e n ta tio n s . In p ra c tic e , how ever, th e y a re u su ally su p p lied differently. T h e aim of th is th e sis is to p ro p o se a lg o rith m s for c o m p u tin g pow er co n ju g ate p re se n ta tio n s for fin ite soluble g ro u p s. T his is achieved in tw o different ways.

O ne of th e w ays in w hich a finite soluble group is often su p p lie d is as a q u o tie n t of a finitely p re s e n te d g ro u p . T h e first p a rt of th e thesis is con cern ed w ith design in g an a lg o rith m to c o m p u te a pow er co n ju g ate p re se n ta tio n for a fin ite soluble g ro u p given in th is way. T h e th e o re tic a l b ack g ro u n d for th e alg o rith m is p ro v id ed a n d its p ra c tic a lity is in v e stig a te d on an im p lem en tatio n .

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C h a p ter 1

In tr o d u c tio n

T h e aim of c o m p u ta tio n a l g ro u p th e o ry is to design alg o rith m s for th e stu d y of g ro u p s a n d to develop th e ir p ra c tic a l im p le m e n ta tio n s . In th is c o n te x t a g ro u p is su p p lie d as in p u t a n d in fo rm a tio n a b o u t th e g ro u p c o n s titu te s th e o u tp u t. T h e re are cases w here it is n o t know n w h e th e r an a lg o rith m exists to c o m p u te ce rta in in fo rm a tio n a b o u t g ro u p s, a n d even cases w h ere it can b e p ro v ed t h a t no a lg o rith m ex ists. F or exam ple, th e re is no alg o rith m to decide w h e th e r a finitely p resen te d g ro u p is finite. In m o st cases o f in te re s t th e ex istence of an a lg o rith m is estab lish ed a n d its p ra c tic a lity becom es th e im p o r ta n t a sp e c t of th e in v e stig atio n . E ven in th o se cases in w hich th e existen ce is n o t g u a ra n te e d , it can u su ally be achieved by in tro d u c in g resou rce re stric tio n s.

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a b ase a n d a s tro n g g en era tin g set. Such an a lg o rith m is, for ex a m p le , d e sc rib e d in Sim s (1970). Secondly th is d escrip tio n is used to in v e stig a te f u r th e r . In th is sense a d e s c rip tio n by a b ase a n d a stro n g g en era tin g set is co n sid ere d to b e a

core d e s c r ip t io n of th e gro u p . In th e design of a lg o rith m s for p e r m u ta tio n g ro u p s it is u su a lly a ssu m e d th a t th e g ro u p is given by a b ase a n d a s tro n g g e n e ra tin g set.

T h is how ever m e a n s th a t it is also im p o rta n t to design alg o rith m s w hich c o m p u te th e core d e sc rip tio n of a g ro u p su pplied in som e o th e r way.

A n im p o r ta n t class of g rou ps are th e polycyclic g ro u p s. T h e y a re c h a ra c te ris e d by th e fa c t th a t th e y have a polycyclic series, w hich is a d escen d in g series of s u b g ro u p s, w h ere each one is n o rm a l in th e p revious one a n d th e successive q u o tie n ts are cyclic (see Segal, 1983, or for c o m p u ta tio n a l a sp ects S im s, to a p p e a r). P olycyclic p re s e n ta tio n s are a n a tu ra l way of d escribing polycyclic g ro u p s as th e y ex h ib it a polycyclic series of th e group . T hey have com e to p lay th e role of th e core d e s c rip tio n for polycyclic gro u p s. T h is is due to th e fact th a t th e y allow th e p ra c tic a l c o m p u ta tio n (by collection) of a n o rm a l w ord for every elem en t in th e g ro u p . As a consequ en ce, it enables th e c o m p u ta tio n of p ro d u c ts a n d inv erses of g ro u p e le m e n ts by calcu latin g a n o rm a l w ord for th e ir fo rm al p ro d u c t or inverse. For an a r b itr a r y polycyclic p re se n ta tio n a g roup elem ent n eed n o t be re p re s e n te d by a u n iq u e n o rm a l w ord. However, it is im p o r ta n t to have a u n iq u e n o rm a l w ord, b eca u se th e n th e p re s e n ta tio n allows th e so lution of th e w ord p ro b le m for th e g ro u p it d escrib e s. A polycyclic p re s e n ta tio n in w hich every g ro u p elem en t is re p re s e n te d by a u n iq u e n o rm a l w ord is called c o n s is te n t. T h e re is a p ra c tic a l a lg o rith m th a t can b e u se d to o b ta in a co n sisten t polycyclic p re s e n ta tio n for a g ro u p given by a n a r b itr a r y polycyclic p re se n ta tio n . A lg orithm s using such d e sc rip tio n s for finite polycyclic g ro u p s are, for exam ple, d escribed in L aue et al. (1984) a n d are now an in te g ra l p a r t of th e c o m p u ta tio n a l g ro u p th e o ry sy stem s C ayley (C a n n o n , 1984) a n d GAP (S c h ö n e rt et al., 1993).

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In th is th esis a tte n tio n is focused on fin ite soluble g ro u p s. Polycyclic p re se n ta tio n s for finite p -g ro u p s , i.e. g ro u p s of p rim e pow er o rd e r, w ere first d escrib ed by Sylow (1872). F or finite soluble g ro u p s th e y a re now kno w n as A G -p re se n ta tio n s, p o w e r-c o m m u ta to r p re s e n ta tio n s or pow er c o n ju g a te p re s e n ta tio n s . H ere th e la st te rm is used. T h e m a in th e m e of th is th esis is to o b ta in pow er c o n ju g a te p re se n ta tio n s for fin ite soluble g ro u p s.

Som e a lg o rith m s for d e te rm in in g co n siste n t pow er c o n ju g a te p re s e n ta tio n s for soluble g ro u p s p re se n te d in a different way alre a d y ex ist. Sim s (1990b) d escribes an a lg o rith m th a t co m p u tes a pow er c o n ju g a te p re s e n ta tio n for a soluble p e rm u ta tio n gro u p . For p -g ro u p s , p ra c tic a l alg o rith m s have b een well e sta b lish e d . For ex am p le, an a lg o rith m now called th e p rim e q u o tie n t a lg o rith m , co m p u tes a pow er c o n ju g ate p re s e n ta tio n for a p-g ro u p describ ed as th e q u o tie n t of a finitely p re se n te d g ro u p to a te rm of th e low er e x p o n e n t-p c e n tra l series (see H avas a n d N ew m an , 1980).

For a long tim e g ro u p th e o rists a n d o th e rs have b een in te re s te d in listin g all g ro u p s of a c e rta in o rd e r u p to iso m o rp h ism . As early as 1854 C ayley listed th e g roups of o rd e r 4 a n d 6 (C ayley, 1854). N u m ero u s classes of g ro u p s w ere listed by h a n d in th e su b se q u e n t cen tu ry , cu lm in a tin g in th e w ork of H all an d Senior (1964), w ho listed all g ro u p s of o rd e r 64. M ore recen tly c o m p u te rs have b een em ployed, for ex am p le by J a m e s , N ew m an an d O ’B rien (1990) to list all g ro u p s of o rd er 128, an d by O ’B rien (1991) to list all g ro u p s of o rd e r 256, by su p p ly in g core d escrip tio n s. T h e y u sed an a lg o rith m , th e p -g ro u p g e n e ra tio n alg o rith m , w hich lists co n siste n t pow er co n ju g a te p re s e n ta tio n s , one for each iso m o rp h ism ty p e of p -g ro u p of a given o rd e r. T h is a lg o rith m is d escrib e d in N ew m an (1977), A scione (1979), a n d O ’B rien (1990). In essence th is m e th o d c o m p u te s for a p -g ro u p G of e x p o n e n t-p class c a list of pow er co n ju g a te p re s e n ta tio n s co n sistin g of one for each iso m o rp h ism ty p e of c e n tra l do w n w ard ex ten sio n by an ele m e n ta ry ab elian p -g ro u p , such th a t th e ex ten sio n h as e x p o n e n t-p class c -f 1 a n d h as its la rg e st q u o tien t of e x p o n e n t-p class c iso m o rp h ic to G.

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soluble g ro u p s w hich are certain do w nw ard ex ten sio n s of a given solu b le g ro u p . T h is n e c e ssita te s th e division of th e thesis in to tw o p a r ts . C h a p te rs 4 a n d 5 are co n c e rn e d w ith solving th e first p ro b lem w hile C h a p te r 7 in v e stig a te s th e second

p ro b le m .

T h e first m e th o d of o b tain in g pow er c o n ju g ate p re s e n ta tio n s for fin ite soluble g ro u p s is th e ta s k of a finite soluble q u o tien t a lg o rith m . A n u m b e r of p ro p o sals

for fin ite soluble q u o tien t alg o rith m s have b een m a d e , for in s ta n c e by W am sley (1977), by L ee d h am -G reen (1984) an d by P lesken (1987). T h e la st a lg o rith m has b e e n d evelo ped, an aly sed an d im p lem en ted by W egner (1992).

H ere we p re s e n t a new finite soluble q u o tie n t a lg o rith m . It h a s m a n y sim i la rities w ith th e p rim e q u o tien t alg o rith m describ ed by H avas a n d N ew m an . In p r e p a ra tio n for th e d escrip tion of th e soluble q u o tie n t a lg o rith m , C h a p te r 3 co n ta in s a b rie f th e o re tic a l d escriptio n of th e ir p rim e q u o tie n t a lg o rith m . I t focuses especially o n th o se th e o re tic al asp ects of th e alg o rith m w hich h av e sign ificant in flu ence on th e p ra c tic a lity of th e soluble q u o tien t a lg o rith m su b se q u e n tly p re se n te d

b u t have n o t b een p u t in to w riting before.

F or a list £ of in teg er p airs C h a p te r 4 in tro d u c e s th e co n ce p t of an £ -s e r i e s, w hich is fu n d a m e n ta l to th e descrip tio n of th e g ro u p s co n sid ered . T h is series is a g e n e ra lisa tio n of th e lower ex p o n e n t-p cen tral series a n d plays th e sam e role in th e soluble q u o tie n t a lg o rith m th a t th e lower e x p o n e n t-p c e n tra l series plays in th e p rim e q u o tie n t a lg o rith m . A gen eralisatio n of th e p -co v erin g g ro u p in th e co n tex t of p -g ro u p s to th e case of soluble group s is th e £ - covering group. A n alg o rith m

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C h a p te r 5 in v e stig ates p ra c tic a l a sp ects of a n im p le m e n ta tio n a n d p o in ts o u t a few im p o r ta n t co n sid era tio n s. W e d escrib e how to m odify fe a tu re s of a n im p le m e n ta tio n of th e p rim e q u o tie n t a lg o rith m for th e c o n te x t of o u r so lub le q u o tie n t alg o rith m a n d in d ic a te in w hich cases th e soluble q u o tie n t a lg o rith m p ro fits fro m th is. W e also p re s e n t a very b rief o u tlin e of th e alg o rith m p ro p o se d by P lesk en , a n d consider its im p le m e n ta tio n by W egner. T h e tw o a p p ro a c h e s are c o m p a re d a n d an a tte m p t is m a d e to id en tify th e s itu a tio n s w here one is m o re a d v an tag eo u s th a n th e o th e r. U n til recen tly th e only available in fo rm a tio n on th e p e rfo rm a n c e of th e im p le m e n ta tio n by W egner w as p ro v id ed in his th esis. U sing his a lg o rith m in th e sam e m a n n e r as we p ro p o se to use th e alg o rith m d escrib e d in th is th esis, significantly b e tte r p e rfo rm a n c e can be achieved. T h is gives a g ood in d ic a tio n as to w here th e inefficiencies of th e a lg o rith m he. F u rth e rm o re , it su g g ests th a t it also co n tain s v aluable tech n iq u es for o b ta in in g a p ra c tic a l soluble q u o tie n t alg o rith m . We discuss th e se p o in ts in d e ta il a n d su ggest th a t a h y b rid of th e tw o a lg o rith m s be used. F u r th e r in v e stig atio n in to th e p e rfo rm a n c e of th e alg o rith m s is necessary.

In C h a p te r 6 an in v e stig atio n of a qu estio n reg ard in g th e freest ex p o n e n t six g roup on tw o g e n e ra to rs pro vides a p ra c tic a l use of th e soluble q u o tie n t alg o rith m .

C h a p te r 7 co n tain s th e th e o re tic a l asp ects of designing an a lg o rith m to list pow er c o n ju g a te p re s e n ta tio n s for all iso m o rp h ism ty p es of th o se soluble g ro u p s on a chosen n u m b e r of g e n e ra to rs w hich are d o w nw ard ex ten sio n s of a given soluble group by c e rta in p -g ro u p s a n d h av e som e ad d itio n a l p ro p e rtie s. T h e m e th o d is a g en era lisatio n of th e p -g ro u p g e n e ra tio n a lg o rith m as d escrib ed in O ’B rien (1990). T h e c h a p te r s ta r ts w ith a very b rie f d e scrip tio n of th is alg o rith m in o rd e r to set th e scene. T h is is followed by a d e sc rip tio n of th e p ro p o sed m e th o d . T h e re m a in d e r of th e c h a p te r c o n tain s th e th e o re tic a l fo u n d a tio n for th e alg o rith m . T h e a lg o rith m has n o t yet b een im p le m e n te d a n d a few h in ts are given as to its im p le m e n ta tio n . T h e alg o rith m requ ires im p le m e n ta tio n to be able to d e te rm in e how well it w orks in p ractice. It is en visag ed th a t a te s t exam p le for it could involve listing pow er co n ju g ate p re s e n ta tio n s , one for each iso m o rp h ism ty p e , of g ro u p s of o rd e r 192.

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£ -covering g ro u p on th is g ro u p . A finite p re s e n ta tio n for a g ro u p w hich h a s S±

as a fa c to r g ro u p is given a n d th e basic step of th e soluble q u o tie n t a lg o rith m is d e m o n s tra te d by co m p u tin g an ep im o rp h ism of th e finitely p re s e n te d g ro u p o n to a fa c to r g ro u p of th e ex ten d ed £ -covering g ro u p of S 4 .

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C h a p t e r 2

B a c k g r o u n d

In th is c h a p te r th e n o ta tio n used in th is thesis is esta b lish e d a n d som e basic resu lts are p re s e n te d .

D e f i n i t i o n 2 .1 For a finite g ro up G th e Frattini subgroup $ ( C ) is th e in te r section of all m a x im a l su b g ro u p s of G.

T h e following p ro p e rtie s of th e F ra ttin i su b g ro u p of a fin ite g ro u p G are well know n a n d can b e fo u n d for in sta n c e in H u p p e rt ( I I I .3, 1967).

• 3>(G) is c h a ra c te ristic in G.

• Let g £ G, th e n g 6 $ ( G ) if an d only if g can be o m itte d from an y g e n e ra tin g set of G.

• Let N be a n o rm a l su b g ro u p of G an d U a s u b g ro u p of G such th a t

N < $ ( U ) , ^ e n N < <L(G).

• Let N b e a n o rm a l su b g ro u p of G, th e n 3>(iV) < <£(G).

• Let N b e a n o rm a l su b g ro u p of G , th e n $ ( G ) N / N < $>(G/N). If N < ^ ( C ) or if G is a p -g ro u p th e n $ ( G ) N / N = $ ( G / N ) .

• If G is a p -g ro u p th e n $ ((7 ) = G ' G 1’.

A g ro u p w hose o rd e r is a pow er of th e p rim e p is a p - g r o u p .T h e following th e o re m , k now n as B u rn s id e ’s B asis T h eo re m , is im p o r ta n t in th e s tu d y of p - g roups (see H u p p e rt, I I I .3.15, 1967).

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D e f i n i t i o n 2 .3 L et G a n d N be finite g ro u p s. A g ro u p H is a downward ext ension of G by N if N is iso m o rp h ic to a n o rm a l s u b g ro u p M of H a n d

H / M is iso m o rp h ic to G.

We refer to a do w nw ard ex ten sio n of G b y N sim p ly as an ex ten sio n of G

by N . N ote th a t th is is n o t th e te rm in o lo g y in use for g ro u p th e o ry in g en eral (see, for ex am p le, Segal, 1983). Since only d o w n w ard ex ten sio n s arise h ere, th is definition should n o t cause any confusion.

D e f i n i t i o n 2 .4 Let G be a g ro u p . D efine a series of su b g ro u p s of G by se ttin g

G (0> = G, G (1) = G ' = ([g,h] I g,h€ G) a n d G (i) = (G ( i- 1 ) )' for i > 0. T h e n

G = G (0) > G (1) > . . .

is a c h a ra c te ristic series of G. T h e g ro u p G is soluble if th e re exists an n £ N such th a t G (n) = {1} a n d th e sm allest such in teg er is th e derived-length of G.

A well know n th e o re m gives us a different c h a ra c te risa tio n of finite soluble g ro u p s (see H u p p e rt, 1.11.9, 1967).

T h e o r e m 2 .5

a) A fi nite group G is soluble if and only if it has a chief series with el eme nt ar y

abelian factors.

b) A fi nite group G is soluble if and only if its composition factors are cyclic

and of p r i m e order.

Let G be a finite soluble g ro u p a n d let

G

=

Go

>

G\ >

• • • >

Gn

= {1} be a co m p o sitio n series for

G

w ith cyclic facto rs of p rim e o rd er. C hoose elem ents a 7; 6

G

for 1 < i < 7i such th a t G,;_i = (G,:,a,;); let p 7; be th e o rd e r of th e facto r

Gi- i/G i.

T h e n A — { a i , . . . , a n } is a g e n e ra tin g set for

G

a n d th e set

R = { a 1-1 = v u , a k’ = v.jk I 1 < i < n , 1 < j < k < n } ,

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O n th e o th e r h a n d , a pow er c o n ju g ate p re s e n ta tio n { A | 7£} of a g ro u p G exhibits

th e c o m p o sitio n series G = Go > G \ > • • • > G n = (1), w here G i - i = (c q ,. . . , a n ) for 1 < i < n . T h e o rd e r of G is a t m o st Ü IL i P* a n d , th e re fo re, G is finite. A w ord , . . . , a n ) in th e g e n e ra to rs is n or m a l if it is of th e fo rm a^1 • • • a w ith 0 < e{ < pi. N o te th a t we only consider w ords in th e g e n e ra to rs A, i.e. th ese w ords do n o t co n ta in inverses of th e g en era to rs.

A n o rm a l w ord u in A is equivalent to a w ord w in A , if u a n d w are th e sam e elem en t of th e g ro u p defined by { A | 7Z}. T h e fu n d a m e n ta l im p o rta n c e of pow er c o n ju g a te p re se n ta tio n s arises from th e o b serv atio n th a t given a w ord w

in th e g e n e ra to rs A, th e pow er co n ju g ate p re s e n ta tio n { A \ 7Z} can be used to c o m p u te a n equ ivalen t n o rm al w ord. T his c o m p u ta tio n of an eq u iv alen t n o rm al w ord is referred to as collection (see e.g. H avas a n d N icholson, 1976, or L eedham - G reen a n d S oicher, 1990). It is assu m ed th a t th e w ords Vjk in a p ow er co n ju g ate p re s e n ta tio n are n o rm al.

If a w ord is n o t n o rm a l it h as a n o n -n o rm a l su b w o rd of th e fo rm a*'* o r of th e form a j a i w ith j > i.

D e f i n i t i o n 2 .6 A collection process relative to the power conjugate presentat ion { A I 7Z} rep laces a w ord w in th e g en era to rs A by an equ iv alent n o rm a l w ord. T h e p ro cess applies th e following step s rep ea ted ly :

1) rep la ce a1-' for 1 < i < n by th e w ord va\

2) rep lace ajCLi for i < j by th e w ord aiVij.

T h e collection process te rm in a te s if no n o n -n o rm a l su b w o rd s re m a in in th e w ord

w.

L et C be a collection process. A pplying one collection ste p of C to a w ord

w re su ltin g in a w ord v is d en o ted by w — v.

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M u ltip lic a tio n of tw o elem ents of G a m o u n ts to c o m p u tin g a n o rm a l w o rd for

th e p r o d u c t, given by c o n c a te n a tio n . G iven a w ord w in A one can find a n o th e r

w ord u in A su ch th a t u re p re se n ts th e inverse of w in G in th e following way.

In itially co n sid er th e fo rm al inverse u o i w. R eplace an o c cu rren c e in u of th e

inverse a f 1 of th e g e n e ra to r ak for som e k by avk 1u, w h ere v is th e fo rm a l

inverse of Vkk a n d re p e a t th is step . T h is process te rm in a te s w ith u being a w ord

in A for th e inv erse of w. T h u s in version of a g ro u p elem en t a m o u n ts firstly to

c o m p u tin g a w ord in A w hich re p re se n ts its inverse a n d secondly to c o m p u tin g

a n o rm a l w ord for th a t w ord. In p ra c tic e th e inverse can b e c o m p u te d in a m o re

efficient way.

In g en era l, th e re m ay exist m a n y n o rm a l w ords re p re se n tin g a given g ro u p

elem en t. If each elem en t is re p re se n te d by a u n iqu e n o rm a l w ord, th e n th e pow er

c o n ju g a te p re s e n ta tio n is consistent. In th is case tw o g ro u p elem en ts are eq u al

only if th e y are re p re s e n te d by th e sam e n o rm a l w ord. For th e fin ite soluble g ro u p

G, given as abo ve, th e o rd e r is eq ual t o f ] ’^ pi.

A ssu m e we are given a collection process relative to a pow er c o n ju g ate p r e

s e n ta tio n { A I 7Z}. A ny collection process obeys th e following rules for w ords

c o n ta in in g p a re n th e se s or b rack e ts. A collection applies a t le ast one collection

step to each su b w o rd in p a re n th e se s before p roceeding. A collection pro cess re

places su b w o rd s in sq u a re b rack e ts by n o rm a l w ords before p ro ceed in g .

T h e following re su lt is due to W am sley (1977). In s u m m a ry it s ta te s th a t a

pow er c o n ju g a te p re s e n ta tio n is co n siste n t if a c e rta in set of w ords in th e g e n e ra to rs

A , th e consistency test words, can b e collected in “sufficiently d iffere n t” w ays a n d

still yield th e sam e n o rm a l w ord.

T h e o r e m 2 .7 Let G be a fi nite soluble group given by the power conjugate

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the f ol lowing words collect to the e m p ty word:

[(«* aj ) a i\ [a fc i aj ai) f ° r 1 < i < j < k < n ,

[ « ) a j ] [ a k'~1 ( a k a j )

[ ( a ^ a ^ a f - 1 ] aj (a]‘)

( a D ai a { (a7-)

- l - l

f o r 1 < j < k < n ,

- l

f o r 1 < i < j < n ,

f o r 1 < i < n.

In im p le m e n tin g an a lg o rith m for deciding w h e th e r a pow er co n ju g a te p re se n ta tio n is c o n siste n t it is m o re efficient to sp lit th e te s t w ords in to tw o w ords, collect each one a n d c o m p are th e resu lt. T h is avoids th e c o m p u ta tio n of inverses. For ex am p le, th e te s t w ord (a k a- ) a ■ a k (a ■ a {)

or a lte rn a tiv e ly th e w ords _{( a k a j ) a i} a n d - l

can be c o m p a re d w ith th e id e n tity

a k (dj a^) can be c o m p a re d . H ere we co n sid er te s t w ords since th is is of a d v a n ta g e for th e th e o re tic a l d escrip tio n .

F or m a th e m a tic a l a n d c o m p u ta tio n a l reaso n s, pow er c o n ju g a te p re se n ta tio n s w ith an a d d e d fe a tu re are considered. Let { A \ 7Z} be a pow er c o n ju g a te p resen ta tio n for a finite soluble g ro u p H w ith A = { a i , .. • , a T*}. Let d be th e m in im al n u m b e r of g e n e ra to rs req u ire d to g e n e ra te H. T h e a d d itio n a l p ro p e rty is th a t th e re exists a d -e le m e n t su b se t X of A such th a t X g e n e ra te s H a n d for each g e n e ra to r aj £ A \ X th e re is a t least one re la tio n of 1Z h av in g aj as th e la st g e n e ra to r on th e rig h t h a n d side a n d o ccu rrin g w ith e x p o n e n t 1. O ne of th ese rela tio n s is lab elled th e definition of aj . T h e p re s e n ta tio n { A \ TZ} w ith th is p ro p e rty is called labelled. If G is a g ro u p w ith g e n e ra tin g set { g \, . . . , g i } a n d r is an e p im o rp h ism of G o n to H, we call r a labelled e p im o rp h ism if each g e n e ra to r

aj £ X o ccu rs as th e la s t g e n e ra to r in th e im ag e u n d e r r of a t le a st one of th e g e n e ra to rs gi of G. O ne of th ese im ages is labelled th e definition of th e g e n e ra to r

aj . If { A I 7Z} is a lab elled pow er c o n ju g a te p re s e n ta tio n for th e g ro u p H a n d r a lab elled ep im o rp h ism fro m G to / / , th e n every g e n e ra to r in A h a s a definition e ith e r as a n im age u n d e r r or as a re la tio n in 7Z.

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on th e sam e side of th e rela tio n as th e g e n e ra to r defined by th is re la tio n . For ex am p le th e following is a labelled co n siste n t pow er c o n ju g ate p re s e n ta tio n for £4, th e s y m m e tric g ro u p on 4 le tters:

{ a, 6, c, d a 2 =: c,

6“ = _63,

ft P _II

ft c1' = : d, c2,

d,L = cd, d i = cd, d d2 }

T h e rela tio n s a 2 = c a n d ch = d a re th e definitions of c a n d d, respectively. N ote th a t th is n o ta tio n im plicitly c h ara cterise s th e set X as th e su b se t of A w hose elem en ts do n o t o ccu r as th e la st elem en t of a rig h t h a n d side of a defin itio n . In th e ex am p le X is th e set {a, 6}.

For a labelled e p im o rp h ism th e sym bo l is also used to specify a d efinition. For ex am p le co nsider th e g ro u p G h av in g th e finite p re s e n ta tio n

•0 , y I z 8, 2/3, ( x ~ l y ) 2, (y x 3y x )2 = z 4}.

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C h a p ter 3

A p rim e q u o tien t a lg o r ith m

In th is c h a p te r a prim e qu o tien t a lg o rith m is d escrib e d . T h e ta s k of a p rim e q u o tie n t alg o rith m is to co m p u te a co n sisten t pow er c o n ju g a te p re s e n ta tio n for a finite q u o tie n t of p rim e pow er o rd er of a finitely p re s e n te d g ro u p . H avas a n d N ew m an (1980) describe such an a lg o rith m . H ere we give a d e sc rip tio n of th e ir a lg o rith m w hich focuses on tho se p a rts w hich are relev an t to th e f u r th e r d iscussion an d have n o t b een p u t in to w riting previously.

D e f i n i t i o n 3 . 1 L et G b e a gro u p a n d p a p r im e . T h e ser ie s

G = 7>f(G) > V l ( G ) >■■■ w ith V f ( G ) = [*?_,(< ?), G] ( i P f - i ( G ) ) ”

for i > 1 is th e lower exponent-p central series of G. If th e re ex ists an in teg er c > 0 su ch th a t V P( G) = (1), th e n G is a p -g ro u p a n d th e sm a lle st such in teg er is called th e e xpone nt- p class of G.

If in a given co n tex t only one prim e p is co n sid ered , it is o m itte d in th e n o ta tio n a n d V f ( G ) is d en o ted by

Vi{G).

In p a rtic u la r th r o u g h o u t th is c h a p te r we a ssu m e th a t p is a p rim e. P er definition V \ ( G ) = G ' G P. If G is a p -g ro u p th e n th e F r a ttin i s u b g ro u p $ ((7 ) of G is V \ ( G ) . Let p d be th e o rd e r of th e F ra ttin i q u o tie n t, th e n d is th e generator number of G (cf. 2.2).

L et G be a finite p -g ro u p of e x p o n e n t-p class c a n d let

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be th e low er e x p o n e n t-p c e n tra l series of G. L et {.A | 7Z} b e a c o rre sp o n d in g pow er c o n ju g a te p re s e n ta tio n . D efine a fu n ctio n u> fro m A in to th e set { 1 , . . . , c} m a p p in g a to t if a £ V t - i ( G ) \ V t (G). T h e n w is a weight f u n c t i o n a n d u;(a) is th e weight of a. T h e elem ents of A satisfy th e following rela tio n s 1Z :

n

a7- = a ^ l't,k\ for 1 < i < n a n d

f t f c € A , u ( t i k ) > u ( a . i )

11

a 1-' = aj P J for 1 < i < j < n ,

w ith 0 < a ( i , j , k ) < p a n d th e p ro d u c t sym b o l in d icate s th a t th e g e n e ra to rs ajfc 6 A o ccu r in in creasin g o rd e r. T h e pow er c o n ju g ate p re s e n ta tio n to g e th e r w ith a w eight fu n ctio n is a weighted power conjugate pr e sentat ion a n d is d e n o te d by { A I 7£, u)}.

A w ord v in th e g e n e ra to rs A of G is an r- word, if it is a w ord in th e

g e n e ra to rs a r , . . . , a n .

T h e co ncept of a labelled pow er co n ju g ate p re s e n ta tio n for p -g ro u p s is refined. H ere we ad d itio n ally assu m e th a t th e definition of a g e n e ra to r £ A is e ith e r of th e fo rm a"-' = a j a ^ or a7- — a*..

T h e p rim e q u o tie n t a lg o rith m tak es as in p u t a p rim e p, a fin ite p re s e n ta tio n

{ g i , . . . , g b I r 1(gu . . . , g b) , . . . , r in(g1, . . . , g b)} for a g ro u p G, a n d an o p tio n a l in teg er c. If th e in te g e r c is su p p lied , th e o u tp u t is a labelled co n siste n t pow er c o n ju g a te p re s e n ta tio n for G / V C{G) a n d a labelled ep im o rp h ism of G o n to th is q u o tie n t. If c is n o t su p p lie d , th e re are tw o possibilities. If th e re is a la rg e st p - q u o tie n t of G th e n th e o u tp u t is a labelled co n sisten t pow er c o n ju g a te p re s e n ta tio n for th is q u o tie n t a n d a labelled e p im o rp h ism of G o n to it. If such a q u o tie n t does

n o t exist th e a lg o rith m does n o t te rm in a te .

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q u o tie n t

G/ Vf i G

) a n d a labelled ep im o rp h ism of

G

o n to th is q u o tie n t h as b een c o m p u te d . Now a labelled co n sisten t pow er c o n ju g a te p re s e n ta tio n for

G/Vi+i(G)

a n d a lab elled ep im o rp h ism of

G

o n to th is q u o tie n t is to b e c o m p u te d . T h e basic step ta k es as in p u t:

1) th e finite p re se n ta tio n for G;

2) a lab elled co n sisten t pow er co n ju g ate p re s e n ta tio n for

K

=

G/ Vi ( G) ’,

3) a labelled ep im o rp h ism

0

:

G-»K.

T h e o u tp u t is:

1) a labelled co n sisten t pow er co n ju g ate p re s e n ta tio n for

H

==

G/Vi+i(G);

2) an ep im o rp h ism <j) :

H-»K-,

3) a labelled ep im o rp h ism r :

G-»H

w ith r</> =

0.

If d u rin g th e basic step it is discovered th a t

Vi(G) = Vi+i(G)

th e n

G/Vi(G)

is th e la rg e st p -q u o tie n t of

G

a n d th e a lg o rith m te rm in a te s re tu rn in g th e consis te n t pow er co n ju g ate p re se n ta tio n for

K

a n d th e e p im o rp h is m

0.

T h e basic step is illu s tra te d by th e following d ia g ra m , w h ere th e in p u t is describ ed on th e rig h t a n d th e o u tp u t is d escrib ed o n th e left.

G

ker 0

D e f i n i t i o n 3 .2 Let G be a finite p -g ro u p w ith g e n e ra to r n u m b e r d a n d ex p o n e n t-p class c. T h e g ro u p H is a descendant of G, if H h as g e n e ra to r n u m b e r

d an d H / V C( H ) is isom orph ic to G. It is an im m ed ia te descendant of G if it has e x p o n e n t-p class c - f 1.

T h e o r e m 3 .3 Let G be a finite p-group with generator n u m b er d and exponent-p class c. There exists a fin ite grouexponent-p G * with generator n u m b er d and exexponent-ponent-exponent-p

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Let F b e th e free group o f rank d and R a norm al su b g ro u p o f F such

th a t F / R = G. D efin e R* : = [ R , F ] R P. A scione (1 9 7 9 ) and O ’B rien (1 9 9 0 ) prove

th e th eo rem by sh ow in g th a t th e group G* : = F / R * has th e required p rop erties.

F u rth er, th e y prove th a t th e iso m o rp h ism ty p e o f F / R * d e p e n d s o n ly on G , n o t

on R.

D e f i n i t i o n 3 .4 G* is called the p - c o ve r i n g group o f G. A p ow er co n ju g ate

p resen ta tio n {.A* | TZ*} for G* is called a p -covering p r e s e n t a t i o n for th e p resen 

ta tio n { A I TZ} o f G , if A. is a su b set o f A* and every relation in TZ* differs from

a relation in TZ only by a word in the elem en ts o f A * \ A .

3.1 A p r im e c o v e r in g a lg o r ith m

T h e task o f a p rim e covering algorithm is to d eterm in e a la b elled con sisten t

pow er c o n ju g a te p resen ta tio n for the prim e covering group o f a fin ite p -group K .

Here w e describ e such an algorith m .

• T h e in p u t o f th e p rim e covering algorithm is a lab elled c o n siste n t pow er co n 

ju g a te p resen ta tio n for a finite p -group K .

• T h e o u tp u t o f th e prim e covering algorithm is a lab elled c o n siste n t pow er

c o n ju g a te p resen ta tio n for th e prim e covering group K * o f K .

Let { A I 7Z, ca} be a w eigh ted labelled con sisten t pow er c o n ju g a te p resen ta 

tion for K . C on sider th o se relation s in 7Z w hich are not d efin itio n s and w h ose left

h and side is eith er a1/ w ith u)(a{) < c or a-' w ith o;(aj) -f- a;(a,;) < c + 1. Let

s be th e n u m b er o f th e se relation s. We ob tain a pow er co n ju g a te p resen ta tio n

{ A I 7Z} for K * as follow s. T h e presen tation is not n ecessarily la b e lled , b u t each

gen erator in A has a d efin itio n . Let A be the set { a i , . . . , a n , a Tl4.l 7 . . . , a n_^.a} ,

w here A = { a i , . . . , a n } , and

1) in itia lise TZ to co n ta in all relation s of 1Z w hich are d efin ition s;

2) m o d ify each n on -d efin in g relation a1/ — v a or a ^ — Vjk o f TZ to read

a 1’ = v n a t or a / 1 = Vj^,at for som e t £ { n + 1 , . . . , n T 5} and add th e

m od ified relation s to TZ, w here different n on -d efin in g rela tio n s are m od ified

by different a t ;

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4) a d d th e rela tio n s a \ = 1 to TZ for n + l < t < n + s.

D efine th e fu n ctio n d? from A in to th e set { 1 , . . . , c + 1} by s e ttin g

( w ( a k ) for 1 < k < n ,

cD(afc) = < a>(a;) + 1 if k > n a n d th e d efin itio n of a k h as left h a n d side a?, I u>(aj) -f if th e d efinition of a k h a s left h a n d side a “*.

E ven th o u g h Cj need n o t be a w eight fu n ctio n , we refer to io ( a k ) for a k € A as

th e weight of a k . N ote th a t u;(a{) = u;(ai) = 1 for 1 < i < d.

L e m m a 3 .5 { A | Ü } is a power conjugate pr es entat ion f o r the p-coveri ng

group G* of G.

D e f i n i t i o n 3 .6 A n elem en t of A \ A is a tail. A p- cove ri ng generator is a ta il th a t is defined by a pow er relatio n or by a c o n ju g a te rela tio n w ith left h a n d side a “* w ith a>(ai) = 1. D en o te th e set of all p -co v erin g g e n e ra to rs by £. A w ord w

in A is r - he a v y if every g e n e ra to r o ccu rrin g in it h as w eight a t le ast r.

L e m m a 3 .7 For each a*. 6 A \ A there exists a relation

a k = W( E ) ,

where W ( £ ) is a word in the elements of £ , which is a consequence of the relations

in TZ.

P r o o f : T h e s ta te m e n t follows by reverse in d u c tio n on u>(ak ) from th e n ex t

lem m a. I

For a c o n ju g ate re la tio n in 1Z w ith left h a n d side x y let V[x,;/] b e a w ord in A

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L e m m a 3 .8 L et 6 £ { 3 , . . . , c -f- 1}. I f fo r each ajt £ -4.\*4 with a>(afc) > b there exists a relation

a k = W { S )

which is a consequence of the relations in 7Z, then there exists a relation

a r = V ( S )

which is a consequence o f the relations in 7Z fo r each a r £ A \ A . with u)(ar ) = 6 — 1. P r o o f : W e refer to th e tails for w hich we know of th e ex isten ce of su ch a rela tio n as bein g o f the right kind.

If a T is defined as a p - th pow er th e n ar £ 8 a n d th e re la tio n a r = ar is of th e re q u ire d form .

A ssum e a r is defined by a rela tio n w ith left h a n d side x u . W e p ro v e th e s ta te m e n t by in d u c tio n on m = u)(u). N ote th a t 2m < 6 — 1.

If m — 1 th e n a r £ E a n d th e relation ar — a r is of th e re q u ire d form . All tails in tro d u c e d by pow er relatio ns are of th e rig h t k in d , th u s we only have to consider tails in tro d u c e d by co n ju g ate relations.

F or Tn > 1 th e g e n e ra to r u in 1Z is eith er defined by a re la tio n w ith left h a n d side y z or w ith left h a n d side y7', because u £ A an d th e p re s e n ta tio n {^4 | 7^.} is labelled. We con sider each case in tu rn .

C ase 1: u is defined as y z . T h e n w[z) = 1 an d th u s c6(y) = m — 1.

L et us co n sid er x y z . In collecting this to a n o rm a l w ord w ith re sp e c t to th e p re s e n ta tio n { A | 7Z} th e re are tw o possibilities to s ta r t. F irs t su p p o se x a n d y

are in te rc h a n g e d . Let C be th e collection process we have chosen.

( x y ) z ^ c

- * c y x i ( ! )

H ere V[x ^y] a n d t[x ,y] are (6 —2)-heavy. T h u s y x plays no f u r th e r role in th e collection pro cess, u n til (1) is rep laced by a w ord of th e fo rm

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In fact, t[x ,y] need n o t be th e la st elem en t in th e w ord listed above; since it is c e n tra l we m ig h t as well assu m e it is a n d we do th is for all o th e r ta ils, to o . H ere

w\ is a n o t n ecessarily n o rm al, (b — 2 ) -h eav y w o rd , since it w as o b ta in e d by

ap plying collection step s to V[x,y)z t[x,y]‘ Since all g e n e ra to rs in V[x,y] are (6 — 2)- heavy, th e only way a co n ju g ate re la tio n can in tro d u c e a ta il of w eight less th a n 6 in to W\ d u rin g th e collection process is by in te rc h a n g in g a g e n e ra to r of w eight 6 — 2 in V[x ^y] w ith z. B u t since u>(z) = 1 th is ta il is of th e rig h t k in d by th e

in d u c tio n h y p o th e sis. T h e collection pro cess co n tin u es :

y x z w \ t ^ x y^ >c y z x v [x,z]i [x,z] W i t [ x ,y]

= y Z X V [ x ^ W l t [ Xi Z] t [ Xf y] .

T h ere are tw o possible directio ns for th e collection p ro cess to co n tin u e. E ith e r

y an d z are in te rc h a n g e d yielding z y u x v ^ x ^ w i t ^ x ^ t ^ x ^ or th e p ro cess w orks on

v [ x , z \ w i before in te rc h a n g in g y a n d z. In th e first case let w 2 — V [ x , z ] w i a n d in

th e second let w 2 be th e resu lt of ap p ly in g a n u m b e r of collection step s to V[ZjZ]tt>i. N ote th a t w 2 is n o t necessarily in n o rm a l form . Since w 1 is (6 — 2 ) -heavy a n d V[x,z] is (6 — m ) - heavy, th e tails in tro d u c e d by c o n ju g ate rela tio n s in ap p ly in g collection steps to vtx ^ w 1 are of th e rig h t kind. I11 e ith e r case th e collection process yields

y z x w 2i[r,,z)i[x,y]- +c z y u x w 2 i [ x ,z]t[x,y]

— z y u x w 3t [x^ t [ x,y].

H ere w3 is n o rm al.

Now we com e back to collecting x y z an d su p p o se th a t a collection p ro cess D

s ta rts by in te rc h a n g in g y an d z.

x ( y z) —* x z y u

*D z xv[x ^ t [ x^ y u

= z xv[ X'Z]yut[X'Z]

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Here w4 and w$ are not necessarily norm al, (b—l) -h e a v y words. All tails of weight

less than b are introduced by conjugate relations involving a generator g in

and y. Since w( g) > b — m and w( y) = m — 1 these tails are of the right kind

by the induction hypothesis. There are two possibilities for the collection process

to continue. Either, it is applied to w±u or to xy. In either case it is applied to

z x y w Gu w 7t[x^z], where w Gu w 7 is obtained in applying a num ber of collection steps

to W4UW5. B ut since w(u) = m and w± is (6 — m ) -heavy, all tails introduced by

conjugate relations by applying collection steps to w±uw$ are of weight at least

b and thus by the induction hypothesis of the right kind. T he collection process

continues:

z x y w cu w 7t[x^ - > D z y x v [ x^ t [ x^ w 6u w 7t[x^

— zyxv^j. w Gu w 7t [x ,y ] t [ z ,z ]

* D* ZyXlLWgt^X y^ ^[x,z] •

T he word wg can only contain tails of the right kind. A tail of the least weight

introduced by a conjugate relation has to result from a conjugate relation with left

hand side g u, where g is a generator in V[ x , y) - T hen w(g) > b—2 and u;(u) = b—m ,

thus the tail is of weight at least b and of the right kind. Thus

* D z y u x v [ x iU] Wg t y x t ^x t [ x ,z ] •

We com pare the norm al words (x y ) z collected by the collection process C and

x [ y z ) collected by the collection process D. Since { A | 7Z} is a consistent presen

tation, th ey differ only in elem ents of A \ A , which are central and of order p. Since

both norm al words begin with z y u x , the norm al words v^x gi[x,u\t[x,y\t[x,z\ and

t[x,z\i[x,y] differ only in elem ents of A \ A . Hence the norm al words v^x^ w g t [ x,u]

and wg differ only in elem ents of .A \.4. Since th ey represent the sam e group el

em ent, we deduce that i[x,u] = w ^ v ^ ^ w g . Thus by expressing each tail on the

right hand side in the elem ents of £ , we obtain a relation expressing i[x,u] as a

word in the elem ents of 8 , which is a consequence of 7Z, since we obtained it by

collection.

Case 2 : u is defined by u := y v . Then w( y ) — m — 1. In collecting the word

x y p there are two different ways to begin a collection, nam ely either to use the

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W e first consid er th e case w h ere we use th e pow er re la tio n . L et C b e th e chosen collection process. T h e n

x { y P)-*C xu

>C t [ x , u ] •

If we collect th e w ord by in te rc h a n g in g x an d y a n d using a collection process D

we o b ta in :

( x y ) y I,- 1 - ^ D y x v [x,y]i [x,y]y p- 1

— y x v [x,y]yI

T h e elem en t x plays no f u rth e r role d u rin g collection u n til a n o th e r y is m oved p a s t all g e n e ra to rs in B u t since a g e n e ra to r in h as w eight a t le ast 6 — 2 a n d y has w eight m — 1, th e tails in tro d u c e d have w eight a t le ast 6 if m > 1 an d are th u s of th e rig h t kind, or are in tro d u c e d by c o n ju g ate re la tio n s in w hich th e second g e n e ra to r h as w eight 1 a n d are th e re fo re of th e rig h t k in d . T h e re are a n u m b e r of possibilities for th e collection process to co n tin u e. E ith e r x a n d y

can be sw a p p ed , or one or m o re o ccu rren ces of y can be m oved p a s t som e of th e g e n e ra to r s in o r th e g e n e ra to rs in tro d u c e d by p re v io u s c o n ju g a te re la tio n s. E ventually, how ever, all o ccu rren c es of y have to be m oved p a s t x , each c re a tin g th e ta il All g e n e ra to rs th a t are in te rc h a n g e d w ith y d u rin g th e collection

process, ex ce p t x, h ave w eight a t le ast 6 — 2 an d in tro d u c e tails of th e rig h t k in d , by th e discussio n above. T h u s we finally o b ta in

y px w t Px ^ = u x w,

w here w is (6 — 2 ) -heavy. N ote th a t t Px ^ is triv ial, since it h as o rd e r p.

A gain, by co m p a rin g th e n o rm a l w ords equivalent to x ( yv ) a n d ( x y ) y p 1, we

deduce th a t t[x ,u] —v [~ u ]1 0 * w hich is a w ord in th e elem ents of *4\.4., w hich are of th e rig h t k in d . T h u s by ex p ressin g each tail on th e rig h t h a n d side in th e elem en ts

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W e o b ta in a p re se n ta tio n { Ä | 7Z} for th e sam e g ro u p by rep lacin g th e o c c u rre n c e of every elem en t in ,4.\£ in every re la tio n in TZ by a c o rresp o n d in g w ord in th e elem en ts of £ a n d settin g A to b e A ö £. In p ra c tic e , th e elem en ts of A \ £

are never in tro d u c e d to th e p re s e n ta tio n , a n d every tim e a w ord in th e elem en ts of £ is c o m p u te d , it is in serted in to th e p re s e n ta tio n im m ed iately .

T h e n ex t le m m a shows th a t th e set of p -co v erin g g e n e ra to rs can be red u ced even fu rth e r. L et T be th e set of th o se elem en ts a k in £ for w hich a k is defined as a c o m m u ta to r or as th e p - th pow er of a g e n e ra to r p, w here p is defined as a p - th pow er.

L e m m a 3 .9 For each a k £ A \ A there exists a relation

a k = W { f ) ,

where W( J - ) is a word in the elem ents o f T , which is a consequence o f the relations in iZ.

P r o o f : T h e s ta te m e n t follows by reverse in d u c tio n on cj(a^) fro m th e n ex t le m m a. T h e in d u c tio n h y p o th e sis, i.e. th e case <F(ak ) = c + 1, is proved by th e sam e a rg u m e n t as used in th e p ro o f of th e n ex t le m m a specialised to this case. |

L e m m a 3 .1 0 Let b £ { 3 , . . . , c + 1}. I f fo r each a k £ A \ A with w( a k ) > b there exists a relation

a k = W ( F )

which is a consequence of the relations in TZ, then there exists a relation

ar = V ( f )

which is a consequence of the relations in IZ fo r each ar £ ^4\^4 with u>(ar ) = 6—1.

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T . A n e lem en t of £ is a tail o f the right ki nd, if we know th a t it c a n b e ex p ressed as a w o rd in th e elem en ts in T .

L et a € £ \ F w ith d>(a) — b - 1. T h e n a is defined as th e p - t h pow er of an e le m e n t u , a n d u is defined by a co n ju g ate rela tio n . L et th e d efin itio n of u have left h a n d side x y . Since u £ A we know th a t u>(y) = 1, th u s u>(x) = 6 — 3, since

u)[u) = 6 — 2. N o te t h a t 6 — 3 > 1.

In collecting th e w ord x v y th e re are tw o different w ays to b eg in a collection, n a m e ly e ith e r to use th e pow er relatio n for x or to in te rc h a n g e x a n d y.

W e first co n sid er th e case in w hich we use th e pow er re la tio n . L et C b e th e collection p ro cess we have chosen. T h e n

( x 1' ) y —>c v xr t xr y

— yV-sJ' ^ x J ' v [ x J’ ,y] t [ x i ‘ ,y]

— 2/Ux j< U [x i< t xv i[ i f ,-«/].

If we collect th e w ord by in terch an g in g x a n d y an d u sing a collection p ro cess

D we o b ta in :

x v ~ l (x y) —>d x v ~ l y x u

—>£> x 1 ~ y x u x u .

T h e re a re tw o possibilities for th e collection process to co n tin u e. E ith e r x a n d

y can b e sw a p p e d , or occu rren ces of x can be m oved p a s t o ccu rren c es of u.

E v en tu ally , how ever, all occu rrences of x have to be m oved p a s t y, w here each m ove c re a te s th e g e n e ra to r u. W h en sw apping x a n d u we in tro d u c e

Since t[x ,v\ h as w eight 6 it is of th e rig h t kind. All g e n e ra to rs o c c u rrin g in V[x ^

have w eigh t a t le ast 6, th u s th e tails in tro d u c e d in collection involving a g e n e ra to r

in V[x ,u\ are ° f th e rig h t kind. T h u s we finally o b ta in

y v xv t xi’Upw = y v xv t xv a w ,

w here w is 6-heavy.

W e d ed u ce th a t a = w ~ 1V[xp ^ t [ xP^ 1 w hich is a w ord in th e elem en ts of A \ A ,

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in th e elem en ts of T , we o b ta in a relatio n ex p ressin g a as a w ord in th e elem en ts of T w hich is a consequence of 77, since we o b ta in e d it by collection. |

We o b ta in a pow er c o n ju g ate p re s e n ta tio n { A \ 77} for th e p -co v erin g g ro u p

G* of G on AVJJ- by replacin g each a* £ A \ J - by th e c o rre sp o n d in g w ord V ( J - )

as given by th e la st lem m a. T h is p re s e n ta tio n is n o t necessarily co n siste n t.

3.2 A p r im e q u o tie n t a lg o r ith m

Let us now r e tu r n to th e p rim e q u o tien t a lg o rith m . T h e ta sk of th e basic ste p is to o b ta in a labelled co n sisten t pow er co n ju g ate p re s e n ta tio n for th e q u o tie n t H

of th e p-co v erin g g ro u p K * of K = G / V i ( G ) a n d a labelled ep im o rp h ism r of G

o n to H . T h is is done using a refined version of T h e o re m 2.7. T h is version, w hich applies to p -g ro u p s , can be fo u n d in V aughan-L ee (1984).

Let Y be th e F?,-sp a c e w ith basis ( a Tl+ i , . . . , a n+ Ä}. Let M d e n o te th e k ernel of th e e p im o rp h ism of H o n to K ; th e n M is a h o m o m o rp h ic im ag e of Y. T h e following th e o re m describes th e kernel of th e h o m o m o rp h ism fro m Y o n to M in a way su ita b le for c o m p u ta tio n . It considers c e rta in n o n -n o rm a l w ords in K * .

T h e o r e m 3 .1 1 Let Y be the free abelian group on { a u_j-i, . . . , a w+ a} of exponent p and { A I 77} the p- cove ri ng presentat ion of the d -generator group G/7^ i( G). Let W consist of the following words in { a i , . . . , a n } :

(a k ai ) ai a k ( a , a i)

. I ' - l

iakaj)

-1

- l

(a3 a i ) a i 1 a , ( a ?)

K K : a i (a D

- 1

f o r l < i < j < k < n ,

f or 1 < j < k < n , j < d,

f or 1 < i < J < n ,

f or 1 < i < n.

Let S be the set of el ements of Y obtained by collecting the words in W with

respect to { A \ 77} and let T = . . . , g l ) \ 1 < i < m } be the set of el ements of Y obtained by evaluating the relators of G in the images of the generators of

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A b asis { m i , . . . , m t } for M can be c o m p u te d using th e G a u s sia n elim in atio n a lg o rith m . L et A* b e th e set A U { m i , . . . , m t }. T h e im ag e o f each elem en t a n_|-i can be e x p ressed in th e b asis of M . T his ex p ressio n rep laces a Tl+i in 1Z a n d in th e e p im o rp h ism . T h u s a la b elle d co n sisten t pow er c o n ju g a te p re s e n ta tio n { A * | 7Z*}

for G / V i + i ( G ) a n d a lab elled ep im o rp h ism r fro m G to H a re o b ta in e d . In p a r tic u la r it can b e show n th a t th e elem en ts in th e b asis of M are w ords in th e elem en ts of T . T h e re fo re , th e fu n ctio n lv in d u ces a w eight fu n c tio n on {*4* I 7Z*} by se ttin g u>*(afc) = u>(a*;) for € A a n d u;*(mjfc) = c + 1.

3 .3

C o n c lu s io n s

Im p le m e n ta tio n s of th e p rim e q u o tien t a lg o rith m exist a n d h ave p ro v ed to be very successful in th e an aly sis of p-q u o tien ts of finitely p re s e n te d g ro u p s.

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C h a p t e r 4

A fin ite s o lu b le q u o tie n t a lg o r ith m

T h e ta sk of a finite soluble q u o tien t a lg o rith m is to c o m p u te a labelled co n

sisten t pow er c o n ju g ate p re s e n ta tio n for a finite soluble g ro u p given as a q u o tie n t of a finitely p re se n te d g ro u p . F irs t th e a lg o rith m is c o m p ared to th e b e tte r kn ow n p rim e q u o tie n t alg o rith m as o u tlin ed in C h a p te r 2. T h is d iscussion also serves as a guide to th e ex p o sitio n of th e a lg o rith m in th e re m a in d e r of th is c h a p te r.

4.1 A c o m p a riso n

T h e p rim e q u o tie n t alg o rith m co m p u tes a labelled co n siste n t pow er c o n ju

g ate p re s e n ta tio n for a p rim e q u o tie n t of a given finitely p re se n te d g ro u p

G

a n d a given p rim e p. It w orks w ith th e lower e x p o n e n t- p c e n tra l series of

G.

It re

p e a ts a basic step w hich, given a labelled co n sisten t pow er co n ju g a te p r e s e n ta

tion of

G/ Vi ( G), c o m p u te s a lab elled co n sisten t pow er c o n ju g a te p re s e n ta tio n of

G/Vi+i(G).

T h e first ta s k of th e basic step is to co m p u te a p re s e n ta tio n for th e

p-covering g ro u p of

G/ Vi ( G), w hich is a c e n tra l ex ten sio n of

G/Vi(G)

by an

e lem en tary ab elian p-g ro u p . T h e p-covering g ro u p of

G/Vi(G)

h as a q u o tie n t

isom orphic to

G/Vi+i(G).

In th e p rim e covering alg o rith m th e G a u ssia n e lim in a tion a lg o rith m is used to c o m p u te a vector space basis for th e e le m e n ta ry ab elian

p -g ro u p by w hich

G/'Pi(G)

is e x te n d e d . A fterw ard s th e kernel of th e p -co v erin g group o n to th e la rg e st q u o tie n t w hich is a h o m o m o rp h ic im ag e of

G

is c o m p u te d .

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sp ace b asis for th is facto r g ro u p of th e ele m e n ta ry a b elian p -g ro u p . T h is yields a lab elled co n siste n t pow er c o n ju g ate p re s e n ta tio n for

G/Vi+i(G).

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4 .2

T h e s o lu b le ^ -se r ie s

D e fin itio n 4 .1 L et G be a g ro u p . L et C — [ ( p i ,c \ ( p t,c * ) ] b e a list of

p airs co n sistin g of a p rim e, p ; , a n d a non -n eg ativ e in teg er, C{, w ith pi ^ P t+i a n d Ci p ositiv e for i < k. For 1 < i < k a n d 0 < j < C{ define th e fist

£i , j = [ ( p i , c i ) , . . . , ( p i - i , C i - i ) , ( p i , j ) ] . Define £ i,o (G ) = G. F or 1 < i < k a n d 1 < < Ci define th e su b g ro u p s

Cij(G) = P?(CiJ.(G))

an d for 1 < i < k define th e su b g ro u p s

C i + l A G ) — £ i , c i ( G )

an d

C{G) = Ci:,Ck(G).

N ote th a t

Cij(G)

>

Cij+i(G)

holds for j < C{.

T h e c h a i n o f s u b g r o u p s

G

=

£ ll0(G ) > C l t l ( G) >■■■> C lfCl( G) = j

C2,

o

(G)

> • • > C k,Ck( G) = C ( G

)

is called th e soluble C- s er i es of G. If C( G) = (1) th e n G is a n C- group. If

C( G) =£ (1) for C = [ ( p i ,c x ) ,. . . ,(pfc,Cfc — 1)] th e n G is a strict C -group.

N ote th a t in th e defin itio n of stric t £ -grou p only th e e x p o n e n t- p k class of

Ci,,o(G) is d e te rm in e d . F or every finite soluble g ro u p G th e re ex ists a n o t n eces sarily u n iq u e fist C su ch th a t G is a stric t £ -g ro u p .

C onsider for ex am p le a soluble g ro u p G w ith £ -s e rie s [(3, 2), (2, 2)] s u c h t h a t

th e o rd e r of C \ ^ ( G ) / C \ ^ ( G ) is 32, th e o rd er of C \ , \ ( G ) / C \ ^ ( G ) is 3, th e o rd e r of £2,o(Gr) / £ 2,i(G ) is 2 10 a n d th e o rd e r of C2, \ { G) / C2,2{G) is 2 55. T h e n th is can

Computing presentations for finite soluble groups