RELATION MODULE S O F FINIT E GROUP S 3
that w e ar e assuming ou r grou p G t o b e finite!) , R/[R, F] is th e direc t su m o f
R n F'/[R, F] an d a fre e abelia n grou p isomorphi c t o F/F'. NO W
(3) d(R/[R, F]) - d(F) = d{R D F'/[R, F])
and th e invarianc e o f thi s numbe r i s reall y a consequenc e o f th e fact tha t R O F'AR, F]
itself i s a structural invariant o f G . Thi s wa s discovere d sevent y year s ag o b y Schu r
[39]. Nowaday s w e kno w thi s grou p t o b e H
2
(G, Z ) (whethe r G i s finit e o r not) .
Note that , becaus e d{R/[R, F]) d
p
(R), equatio n (3 ) show s th e number in Questio n
3 i s alway s nonnegative .
Schur's interes t i n thes e matter s aros e fro m th e following . Give n a representatio n
of G a s a grou p o f collineation s o f a projective geometr y ove r C , sa y y : G PGLO2 , C),
and recallin g tha t
1 - C * GL(« , C ) - PGL(T2 , C ) 1
is exac t (wher e C i s th e multiplicativ e grou p o f C) , Schu r aske d i f on e coul d con -
struct a centra l extensio n b y G o f th e for m (2 ) an d a linea r representatio n rj: E —•
GL(n, C ) s o tha t rj induce s y ; i n othe r words , s o tha t
E G
GL(*2, C ) PGL(T2 , C )
is commutative . Schu r prove d tha t thi s is indee d th e case . Th e group s E arisin g i n
this resul t ar e no w calle d th e Schur covering groups o f G . The y hav e K ~ HAG, Z) .
Thus K i s wha t G mus t b e "multiplie d by " t o giv e E (an d thi s i s wh y Schu r calle d
K th e multiplicator o f G).
1. 6 Suppos e nex t tha t th e kerne l K i n ou r extensio n (2 ) i s n o longe r centra l bu t
is abelian . The n K i s a G-module : w e se t kg = e" ke, wheneve r k 6 K an d ed = g.
It follow s tha t Ke r f R 1 = [R, R] an d (2 ) i s a n imag e o f th e free abelian extension
([18, §9.5] )
1-+ R -*F-+G 1,
where bar s denot e thing s modul o R . Thi s i s th e situatio n w e shal l stud y i n thes e
Lectures. Th e G-modul e R i s calle d th e relation module determine d b y the presentatio n
(1).
Since th e additiv e structur e o f R i s a fre e abelia n group , R i s wha t i s calle d a
TjG4attice. I t i s th e representatio n theor y o f suc h lattice s a s i t ha s bee n develope d
during th e las t twent y year s tha t wil l b e ou r principal tool .
Question 3 has a n affirmativ e answe r here : d
p
{R) - d(F) i s a n invarian t fo r G .
Hence T{G), th e minimu m o f th e number s dAR) = dAR) fo r al l presentation s (1), i s
realised b y even / minima l presentatio n [19]. (Cf . §7. 3 below. )
Notice that , b y (3) ,
Previous Page Next Page