Chapter 0 . Basi c notion s an d result s
This brie f sectio n contain s som e materia l whic h i s basi c i n th e sequel . Mos t
proofs i n thi s sectio n wil l eithe r b e ver y muc h abbreviate d o r omitte d altogether . Com -
plete proof s ca n eithe r b e furnishe d directl y o r ca n b e extracte d fro m th e fundamenta l
paper b y P . Hal l [ 2 9 ], o r fro m th e boo k b y M . Hall , Jr . [ 2 7 ], o r obtaine d fro m th e
references cited .
0.1. Le t X b e a propert y (o r class ) o f groups . A finit e norma l serie s (i.e. , eac h
term i n th e serie s i s a norma l subgrou p o f th e succeedin g term )
1 = G o G i ---G
t
= G (1)
of G i s terme d a poly-X series for G i f G . , / C X fo r i - 0 , , / - 1. A grou p G
is terme d poly-X i f i t ha s a t leas t on e poly- ^ series . Th e length o f th e serie s (1) i s
I
Polyabelian group s ar e usuall y referre d t o a s solvable.
The cente r o f a grou p G i s denote d b y CG. A n invarian t serie s (1) (i.e. , G . i s a
normal subgrou p o f G fo r i = 1, , / ) i s terme d a central series i f G. ./G .
£(G/G .) fo r i - 0 , « - ,/ 1. G i s nilpotent i f i t ha s a t leas t on e centra l series ; th e
class o f a nilpoten t grou p i s th e leas t o f th e length s o f it s centra l series .
Finitely generate d nilpoten t group s ar e polycycli c (thi s ca n b e prove d b y observ -
ing tha t i f G i s a finitel y generate d nilpoten t grou p o f clas s c the n A . G i s als o
finitely generate d (se e 0. 2 below)) .
An extensio n o f a finitel y presente d grou p b y anothe r finitel y presente d grou p i s
finitely presented . Henc e poly-finit e ly=presented coincide s wit h finitel y presented . I n
particular then , polycycli c group s ar e finitel y presented .
A grou p G satisfie s th e maximal condition i f i t ha s n o infinit e strictl y increasin g
chain o f subgroups ; thi s i s equivalen t t o th e conditio n tha t al l subgroup s o f G ar e
finitely generated . A n extensio n o f a grou p wit h maxima l conditio n b y anothe r suc h
group satisfie s th e maxima l condition . Thu s th e poly-maximal-conditio n coincide s
with th e maxima l condition . I n particula r polycycli c group s satisf y th e maxima l con -
dition.
Let p b e an y prim e an d le t G b e a nilpoten t group . The n th e element s o f orde r
a power-of- p constitut e a norma l subgrou p o f G, th e Sylo w p-subgrou p o f G.
If G i s an y grou p the n G denote s th e commutato r o r derive d grou p o f G. A
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