Other Monograph s i n thi s Serie s

No. 1 . Irvin g Kaplansky : Algebraic and analytic aspects of operator algebras.

3. Lawrenc e Markus : Lectures in Differentiable Dynamics.

4. H . S . M . Coxeter : Twisted honeycombs.

5. Georg e W . Whitehead: Recent advances in homotopy theory.

6. Walte r Rudin : Lectures on the edge-of-the wedge theorem.

7. Yoz o Matsushima : Holomorphic vector fields on compact Kdhler manifolds.

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Conference Boar d o f the Mathematica l Science s

REGIONAL CONFERENCE SERIES IN MATHEMATICS

supported b y the

National Scienc e Foundatio n

Number 2

LECTURE NOTE S ON NILPOTENT GROUP S

by

Gilbert Baumsla g

Published fo r the

Conference Boar d o f th e Mathematica l Science s

by the

American Mathematica l Societ y

Providence, Rhod e Islan d

http://dx.doi.org/10.1090/cbms/002

Expos i tory Lecture s

from th e CBM S Regiona l Conferenc e

held a t th e Univers i t y o f T e x a s , Aust in , T e x a s

May 2 6 - 3 0 , 196 9

International Standar d Boo k Numbe r 0-8218-1650- 0

Library o f Congres s Catalo g Numbe r 78-14563 6

AMS 197 0 Subjec t C l a s s i f i c a t i o n s : Primar y 2 0 E 15 ;

Secondary 2 0 E 3 5 , 2 2 E 2 5 , 2 2 E 6 0 .

Key word s an d phrases : Ni lpoten t group , algorithmi c problems ,

res idual ly f inite , commutato r c a l c u l u s , l i e an d a s s o c i a t i v e rin g

t echn iques , l i e groups , matri x groups .

AMS-on-Demand ISBN 978-0-8218-4157-0

Copyright O 197 1 b y th e America n Mathematica l Societ y

Printed i n th e Unite d State s o f Americ a

All rights reserved except those granted to the United States Government

May not be reproduced in any form without permission of the publishers.

CONTENTS

Introduction v i

0. Basi c notion s an d result s 1

1. Algorithmi c problem s fo r finitel y generate d nilpoten t group s 5

2. Residua l propertie s an d som e application s 1 0

3- Li e an d associat iv e rin g technique s an d th e commutato r calculu s ,...2 5

4. Li e grou p technique s 3 9

5. Miscellaneou s topic s 6 4

Bibliography 6 8

v

Introduction

These lecture s ar e concerned , i n th e main , wit h finitel y generate d nilpoten t groups .

The theor y o f thes e group s i s ric h an d exciting .

There see m t o b e thre e mai n part s t o thi s theory . Th e firs t o f thes e deal s wit h th e

so-called commutato r calculus , whic h wa s initiate d b y Phili p Hal l i n hi s fundamenta l

paper [ 2 9 ] . A s th e nam e suggest s i t i s concerne d wit h manipulation s o f commutator s

and deduction s o f furthe r relationship s betwee n commutator s fro m basi c identities .

There doe s no t see m t o b e an y guidin g principl e i n thi s calculus ; consequentl y thi s

aspect o f th e theor y i s fo r th e mos t par t rathe r difficult .

The secon d aspec t o f th e theor y is , i n a sense , governe d b y a singl e principle .

This principl e ma y b e likene d t o a well-know n procedur e i n elementar y numbe r theor y

where on e show s tha t a propositio n abou t th e integer s hold s modul o eac h prim e p an d

thence fo r th e integer s themselves . Thi s notio n ma y b e used , i n particular , t o prov e

certain result s abou t finitel y generate d torsion-fre e nilpoten t groups . Th e ke y fac t i s

the followin g theore m o f K . W . Gruenberg [ 2 5 ] : i f G i s a finitel y generate d torsion -

free nilpoten t grou p an d p i s an y prime , then , give n an y elemen t g G G {g /- l ) , ther e

is a norma l subgrou p / V o f G suc h tha t G/N i s a finit e p-grou p wit h g $ N. Roughl y

speaking, th e ide a i s t o sho w tha t i f a propositio n abou t a finitel y generate d torsion -

free nilpoten t grou p hold s fo r al l it s homomorphi c image s o f prime=powe r order , the n i t

holds als o fo r th e grou p itself .

The thir d par t o f nilpoten t grou p theor y stem s i n th e mai n fro m th e connectio n

between li e group s an d li e algebras ; i t wa s discusse d firs t b y A . I . Ma i ce v i n hi s

beautiful pape r [ 6 3 ] . Th e impac t o f thi s connectio n an d th e consequen t connection s

between arithmeti c an d algebrai c group s ha s onl y ver y recentl y emerge d (see , fo r ex -

ample, th e ver y dee p paper s b y L . Auslande r [ 2 ] , [3 ] an d th e pape r b y L . Auslande r

and G . Baumsla g [ 4]). I n a sens e thi s i s th e mos t exciting , althoug h i t i s i n som e

ways th e mos t limited , aspec t o f th e whol e theory . Her e w e shal l develo p ab initio,

by usin g th e approac h o f S . A . Jenning s [ 4 7L a s muc h o f th e necessar y machiner y a s

is neede d fo r ou r discussio n o f th e automorphis m group s o f finitel y generate d nilpoten t

groups.

Although I have chose n t o divid e th e theor y o f finitel y generate d nilpoten t group s

into thre e parts , i t shoul d b e pointe d ou t tha t thes e part s ar e reall y ver y muc h inter -

related, eac h complementin g th e others .

The progra m fo r thes e lecture s i s se t fort h i n th e tabl e o f contents . I have no t

VI

GILBERT BAUMSLA G vn

divided th e materia l u p int o th e thre e part s describe d abov e becaus e thi s woul d mak e

for ver y awkwar d exposition . I hav e a ls o no t trie d t o provid e a complet e accoun t o f th e

theory here . Th e intereste d reade r wil l fin d tha t a stud y o f th e bibliograph y include d

at th e en d o f thes e lecture s wil l allo w hi m t o g o mor e deepl y int o thos e aspec t s o f th e

theory whic h appea l t o hi m most .

Acknowledgement

These note s ar e substantial l y th e sam e a s thos e prepare d a s a n ai d t o th e te n

lectures o n finitel y generate d nilpoten t group s whic h I gav e i n Austi n a t th e Universit y

of Texa s durin g Ma y o f 1969 - The y hav e bee n slightl y polishe d an d numerou s mistake s

have bee n eradicated , mainl y du e t o th e diligenc e o f Joh n F . Ledli e t o who m I woul d

like t o expres s m y appreciatio n fo r hi s valuabl e hel p an d ass i s t ance . I would a l s o lik e

to expres s m y thank s t o Nanc y Singleto n an d Kath y Vigi l fo r thei r patien t decipherin g

of man y rewritten , hal f legibl e page s whic h mad e u p thi s manuscript . Finall y I woul d

like t o than k th e Mathematic s Departmen t (especiall y Joh n R . Durbin ) o f th e Universit y

of Texa s fo r makin g i t suc h a pleasur e t o spen d a wee k i n Ma y i n Austin .

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68 GILBERT BAUMSLA G

Theorem 5.10 . Every solvable group of matrices over the ring of integers of an

algebraic number field is polycyclic.

Theorem 5.1 0 yield s tw o furthe r theorem s o f A . I . Ma i ce v [62 ] namely :

Theorem 5.11 . The solvable subgroups of the group of automorphisms of a poly-

cyclic group are polycyclic.

Theorem 5.12 . A solvable group is polycyclic if and only if all its abelian sub-

groups are finitely generated.

We tur n ou r attentio n no w t o a rathe r differen t notio n du e t o J . Milno r [67 ] whic h

essent ial ly single s ou t th e nilpoten t group s fro m othe r solvabl e groups . Milnor' s

notion i s tha t o f growt h function . I n orde r t o explai n le t G b e a grou p wit h a give n

finite se t X - \x , • • • , x \ o f generators . Fo r eac h positiv e intege r n le t g(n) denot e

the numbe r o f element s o f G tha t ca n b e expresse d a s word s o f lengt h a t mos t n. A s

it stand s g(n) i s a functio n o f th e positiv e intege r n. Th e functio n g i s calle d a

growth function fo r G. W e ter m g o f polynomia l typ e o f degre e < e i f g(n) < cn e

where c i s som e constant . I f g(n) > uv n wher e bot h u an d v ar e constants , v > 1 ,

then g i s sai d t o b e o f exponentia l type . I t turn s ou t tha t i f an y growt h functio n o f G

is o f a give n type , the n the y al l a re , i .e . , th e choic e o f th e finit e syste m o f generator s

of G doe s no t affec t th e typ e o f th e associate d growt h function . Th e relevanc e o f

these notion s t o u s lie s i n th e followin g resul t du e i n par t t o J . Milno r an d i n par t t o

J . A . Wol f (se e [87]) .

Theorem 5-13 - A finitely generated solvable group is either of exponential type or

of polynomial type. It is of polynomial type if and only if it is a finite extension of a

nilpotent group.

The relevanc e o f thi s notio n o f a growt h functio n t o algebr a i s no t clear , bu t i t i s

s t i l l quit e fascinating . I t aros e i n connectio n wit h th e stud y o f curvatur e o f certai n

Riemannian manifold s (se e J . Milno r [68 ] an d J . A . Wol f [87]) .

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