4 K.W . GRUENBERG
dG(R)-d(F)d(H2(G, Z) )
and o f cours e
(4) d
F
(R) d
G
(R) d(R/[R, F]) .
The secon d inequalit y her e ca n easil y b e strict : e.g . [44], § 2 . Bu t whe n G i s a p-
group i t mus t b e a n equality : [ 19] o r [46], Whethe r th e firs t inequalit y i n (4 ) ca n eve r
be stric t remain s a n ope n problem . Wamsle y ha s investigate d thi s questio n extensivel y
for p-groups: cf . [47] .
I •/ Ou r stud y o f th e relatio n module s o f a fixe d finit e grou p wil l fal l unde r thre e
main headings :
(1) th e compariso n o f differen t relatio n modules ;
(2) arithmeti c generatio n properties ;
(3) decompositio n properties .
These thre e topic s are , o f course , highl y interconnecte d an d represen t bu t a firs t
approach t o th e modul e theoreti c aspect s o f presentatio n theory . Th e grou p theoreti c
aspects li e o n a somewha t deepe r level . T o stud y F/R an d it s images , cohomolog y
necessarily enter s th e picture . W e shal l discus s thes e matter s a littl e i n th e las t
lecture.
We now highligh t som e o f th e mai n results . Her e G i s a give n finit e group .
(A) I f Rj , R
2
ar e relatio n group s i n th e sam e fre e grou p F , the n Rj , R
2
ar e
locally isomorphi c a s G-modules : i.e. , i f Z , denote s th e loca l rin g a t p,
Rr ® Z
( }
& R
2
® Z
(p)
fo r al l prime s p .
If d{F) d(G), the n w e eve n hav e R j ^ R
2
: [49] . I t i s stil l a n ope n proble m
whether thi s alway s remain s tru e whe n d(F) = d(G). (Fo r thes e result s cf . Lectur e 5. )
(B) Le t R - A © P , wher e P i s a ZG-projectiv e modul e an d A ha s n o projectiv e
direct summand . The n A i s uniqu e t o withi n loca l isomorphis m whateve r relatio n
module R i s use d an d whateve r decompositio n i s taken . W e cal l A a relation core
for G .
By a basi c resul t o f Swa n (cf . Lectur e 4) , P ® Q ^ (QG) m. If , i n particular, d(F) =
d(G), m turn s ou t t o b e independen t o f th e choic e o f R i n F an d o f th e decomposition .
We cal l 77 2 th e presentation rank o f G an d writ e m = pr(G).
All solubl e group s hav e zer o presentatio n rank . Bu t ther e exis t group s wit h arbi -
trarily larg e presentatio n rank . (Cf . Lecture s 5 an d 6. )
(C) Th e minimu m numbe r o f generator s o f a relatio n modul e ca n b e calculate d
from a suitabl e finit e image . Mor e precisely , ther e exist s a prime p occurrin g i n \G\
so tha t dAR) = dAR/pR). Notic e that , i f G i s a p-group, thi s immediatel y implie s
that d
G
(R) = d(R/[R, F]) .
There ar e simila r theorem s fo r relation core s an d for the augmentatio n idea l g (whic h i s
the kernel o f th e homomorphism Z G —» Z determine d b y g h- » 1 fo r all g i n G) . (Cf . Lectur e 7. )
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