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 [ 47L 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
Previous Page Next Page