T h is c h a p te r describes an alg o rith m th a t c o m p u tes q u o tie n t sy stem s for cer ta in n ilp o te n t fa c to r groups of a g ro u p given by a fin ite p re s e n ta tio n . T h e b a sic a lg o rith m is a re p e a te d ap p lic a tio n of th e m e th o d s in C h a p te r 4. T h e m a in p a r t of th e c h a p te r is co ncerned w ith defining special polycyclic p re s e n ta tio n s for n ilp o te n t g rou ps a n d d escribing an im p ro ved versio n of th e b asic a lg o rith m . T h e m e th o d s u sed for th is alg o rith m are a g e n e ra liz a tio n of th e m e th o d s em ployed by H avas a n d N ew m an (1980) for th e c o m p u ta tio n of w h a t th e y call weighted p o w e r - c o m m u t a t o r presentat ions for fa c to r p -g ro u p s of fin itely p re s e n te d grou p s. T h e first im p le m e n ta tio n of a n ilp o te n t q u o tie n t a lg o rith m h as b een done by C .C . Sim s in M a th e m a tic a .
5.1 The basic algorithm
T h ro u g h o u t th is c h a p te r let ( X , TV} be a fin ite p re s e n ta tio n for a g ro u p G. Let
Fx b e th e free g ro u p on X an d N th e n o rm a l closure of 71 in Fx . F u rth e rm o re , define K = N l c +1 (FX ) a n d L = N [ K , Fx ). T h e n G / 7c+1(G?) = Fx / K . F rom
th is we get L = N [ N - / C+1(FX), Fx ] = N [ N , ][7 c+1( ^ ), Fx ] = N - , C+2(FX ).
T h e second eq u ality follows fro m H ilfsatz III.1 .1 0 a) in H u p p e rt (1967). T h is shows th a t Fx /L = G/ 7 c+2(Cr). If A c is a co n siste n t q u o tien t sy ste m re p re se n tin g
G/ 7c+i(G ), th e n th e c e n tra l covering sy stem Ac+i of A c re p re se n ts Fx / L by th e re m a rk s before T h e o re m 4.4 a n d th e re fo re G / 7 C+2(G ) by th e above a rg u m e n t.
Let C be a p o sitiv e integ er. T h e aim is to c o m p u te a co n siste n t q u o tie n t sy ste m b ased on (X , 7Z) rep resen tin g G/7c+ i(C ?). T h is can be do n e by s ta r tin g
sy ste m A c re p re se n ts (j/7c+1((?), th e n its c e n tra l covering sy ste m *4c+ i re p re
sen ts G/ j c + 2 - A co n sisten t q u o tie n t sy stem for G / j c + i ( G ) can b e c o m p u te d by successively c o m p u tin g A c+i from A c u n til c -f 1 = C.
T h e m a in d raw b ack of th is ap p ro ac h is th a t in co m p u tin g A c th e n u m b e r of new g e n e ra to rs quickly becom es to o larg e for p ra c tic a l p u rp o se s. Let n c be th e n u m b e r of g e n e ra to rs of th e polycyclic p re s e n ta tio n in A c . E ach of th e g e n e ra to rs of th e polycyclic p re s e n ta tio n of A c is defined e ith e r by an im ag e of an elem ent in X or by a polycyclic rela tio n . Let I c b e th e set of in d ices of g e n e ra to rs w ith a pow er re la tio n in th e polycyclic p re s e n ta tio n of A c. T h e re are |X | im ag es of g e n e ra to rs in X , th e n u m b e r of pow er re la tio n s is \IC\ a n d th e n u m b e r of p o sitiv e a n d inverse c o n ju g ate re la tio n s is a t m o st n c( n c —1). T h e n u m b e r of new g e n e ra to rs th a t have to be defined is th e su m of th e se th re e values m in u s th e n u m b e r of g e n e ra to rs b eca u se each g e n e ra to r h as a definition . T h ere fo re th e n u m b e r o f new g e n e ra to rs is a t m o st |X | + \IC \T n c( n c — 2). In o th e r w ords, th e n u m b e r o f new g e n e ra to rs is a q u a d ra tic fu n c tio n of th e n u m b e r of g e n e ra to rs in th e polycyclic p re s e n ta tio n . H ow ever, by T h e o re m 2.3.7 we know th a t 'yc( G ) / /yc+ i ( G ) can be g e n e ra te d by rii e g e n e ra to rs, w here rij is th e n u m b e r of g e n e ra to rs for G/ ~f \ ( G)
a n d e th e n u m b e r of g e n e ra to rs for 7c_i( G ) /7cG. So it is d esirab le to red u ce th e n u m b e r of new ly defined g en era to rs. We will d escrib e an a lg o rith m th a t defines
In S ectio n 2.3 we saw th a t n ilp o te n t polycyclic g ro u p s can b e defined by a special ty p e of polycyclic p re se n ta tio n . T h e s tr u c tu re of th o se p re s e n ta tio n s m akes it possible to c a rry o u t c o m p u ta tio n s w ith n ilp o te n t g ro u p s m ore efficiently th a n w ith gen eral polycyclic p re se n ta tio n s.
w ith th e triv ia l q u o tie n t sy stem Aq b a se d on (X ,7 £ ). If th e co n siste n t q u o tie n t
new g e n e ra to rs.
5.2.1 Definition A polycyclic p re s e n ta tio n is called a nil potent (polycyclic) p r es e nt at i on if th e co n ju g ate re la tio n s ta k e th e follow ing form .
For each p a ir z,
j
£ { 1 ,. .. , n } ,i < j,
th e re are th e re la tio n sa ~ 1aJa t = a j w f + ,
a {a j a ~ l = a j W ^ + , i ^ / ,
a “ 1a “ 1a t = a ~ l w f ~ , j g l ,
a i a j ' a ^ 1 = i , j £ I .
T h e rig h t h a n d sides u>^+ , a n d are w ords in { a J+ 1 , . . . , a n }.
5.2.2 Theorem A g ro up given by a n ilp o te n t polycyclic p re s e n ta tio n is n ilp o te n t.
P ro o f: Let (A,»S) be a n ilp o te n t polycyclic p re s e n ta tio n a n d define
G{
to be th e g ro u p g e n e ra te d by th e elem ents { a * ,. . . , a n }. S im ilarly to th e p ro o f of L em m a 3.1.2 we see th a tGi
is n o rm al inG = G\.
F ro m th e p re s e n ta tio n it is clear th a ta{Gi
is c e n tra l inG/Gi+\.
T h erefo re,G — G\ > .
= 1 is a c e n tra l seriesof
G
a n dG
is n ilp o te n t. ■In S ection 2.3 we defined a w eight fu n c tio n in te rm s of th e low er ce n tra l series of a g ro u p a n d showed th a t th e w eight of a c o m m u ta to r is a t le a st th e su m of th e w eights of th e tw o co m p o n en ts of th e c o m m u ta to r. T h is m o tiv a te s th e follow ing definition.
5.2.3 Definition A weighted nilpotent p r es e nt at i on is a trip le (A,c>,u>) such th a t (A,<S) is a n ilp o te n t polycyclic p re s e n ta tio n in w hich th e rig h t h a n d sides of all re la tio n s are red u ce d w ords a n d w : { a 1?.. . , an } —> N is a fu n c tio n such th a t th e re la tio n s satisfy th e following co n d itio n s
w( a ) > w( a( ) for each g e n e ra to r a o ccu rrin g in
w a a n d w ~{ ,
T h e value w ( a) for a g e n e ra to r a is called the weight of a. T h e w eight of th e inverse of a g e n e ra to r is defined to be th e sam e as th e w eight of th e g e n e ra to r. T h e w eight w ( u ) of a n o n -em p ty w ord u in th e g e n e ra to rs O i, . . . , a n is defined as th e m in im u m of th e w eights of th e g e n e ra to rs o ccu rrin g in u. T h e e m p ty w ord is assign ed th e w eight oo.
By L em m a 2.3.3, th e w eight fu n c tio n for n ilp o te n t g ro u p s defined in S ection
2.3 is a w eight fu n c tio n in th e sense of th e p rev io u s defin itio n . T h e w eight fu n ctio n defined h ere does n o t in d ic a te in w hich te rm o f th e low er c e n tra l series a group elem ent lies b u t r a th e r w hich c o n ju g ate re la tio n s are involved in collectin g a w ord to n o rm a l form . Let (A , S , w) be a w eighted polycyclic p re s e n ta tio n a n d c th e h ig h e st w eight assigned to a g e n e ra to r by th e w eight fu n c tio n w. T h e n tw o g e n e ra to rs ev id en tly co m m u te if th e su m of th e ir w eights is b ig g er th a n c. C o n seq u ently, tw o w ords in th e g e n e ra tin g set co m m u te if th e s u m of th e ir w eights is la rg e r th a n c.
5.2.4 Example T h e following is a co n sisten t w eighted n ilp o te n t p re s e n ta tio n ,
( ° i , Ö2, a-3, a 4 ö j 1 a.2 a i — : a 2 5 ß j *0. 3 (1 1 <230. 4, CL2 * 0 3 0 2 — 0 3 0 4 , ß j 0 4 0 1 — 0 4 , 0-2 &4CL2 — 0 4 , Ö3 0 4 0 3 — 0 4 , ß l ß 2 ß j — 0 2 0 3 * 0 4 , - 1 _ -1 -1 _ -1 ß l ß ß ß i — Ö.3 ^ 4 ? ß 2 &3 &2 — ^ 3 ^ 4 5 ß l ß 4 ß j = 0 4 , 0 2 0 4 0 2 — 0 4 , 0 3 0 4 0 ^ = O4 )
T h e w eight of Oi a n d a 2 is 1, th e w eight o3 is 2 a n d th e w eight of o4 is 3. T h e w ord 0403010^”* h as w eight 1. It can be collected in tw o different w ays. If O ia ^ 1 is rep la ced by th e e m p ty w ord, we get 0403. B ecause 10(03)4-20(04) > 3, th e se two g e n e ra to rs co m m u te a n d we get 0304. T h ere fo re , th e g ro u p elem en t co rresp o n d in g to O3O4, a n d 0403010^*, has w eight 2. H ow ever, it is p o ssib le th a t in collecting th e w ord 0403010^* to n o rm a l form c o n ju g ate re la tio n s are u sed th a t involve g e n e ra to rs of w eight sm aller th a n 2. If th e first ele m e n ta ry collection step applied to 04030!aj-1 is to m ove a! p a s t o 3, th e c o n ju g a te re la tio n a ” 1 a 3a i 0304 is u sed . Since th e p re s e n ta tio n above is co n siste n t, th is will lead to th e sam e red u ced
w ord. T h is ex am p le shows th a t th e w eight of a w ord in d ic a te s w hich c o n ju g ate re la tio n s m ig h t b e u sed in collecting a w ord. If a w ord h as w eight c, th e n c o n ju g ate re la tio n s in w hich th e c o n ju g a tin g g e n e ra to r h as w eight c m ig h t have to be used to collect th e w ord to n o rm al form . O n th e o th e r h a n d , co n ju g a te re la tio n s in w hich th e c o n ju g a tin g g e n e ra to r has w eight sm aller th a n c will n o t be u sed in collecting th e w ord to re d u c e d form .
5.2.5 Lemma Let ( A , S , w ) be a w eighted polycyclic p re s e n ta tio n s w ith g en e ra tin g sequence ( a j , . . . , a n ) a n d let c be th e h ig h e st w eight of a g e n e ra to r in A.
L et i be an in teg er such th a t 1 < i < n a n d w ( a j) -f u>(a*.) > c for i < j , k < n.
If we define A 1 to be th e sequence ( a ; , . . . , a n ), S ' to be th e set o f re la tio n s in S
th a t only involve th e g e n e ra to rs in A ' a n d w ' to be th e re s tric tio n of w to A \