Preface
This bookle t reproduce s a cours e o f te n lecture s tha t I gave a t a n N. S . F . Regiona l
Conference a t th e Universit y o f Wisconsin-Parksid e i n Jul y 1974. Th e ai m o f th e lec -
tures wa s twofold . O n th e on e hand , I wanted t o sho w grou p theorist s ho w th e presen -
tation theor y o f finit e group s ca n nowaday s b e successfull y approache d wit h th e hel p
of integra l representatio n theory . O n th e othe r hand , I hoped t o persuad e rin g theorist s
that her e wa s a n are a o f grou p theor y wel l suite d t o application s o f integra l represen -
tation theory . A s a result , th e cours e ha d t o b e constructe d s o tha t onl y a modicu m o f
either grou p theor y o r modul e theor y woul d b e presuppose d o f th e audience .
The ai m o f thi s printe d versio n remain s th e same . T o achiev e it , I have fel t i t
advisable t o fil l i n an d expan d th e Parksid e lecture s a t a numbe r o f places . Fo r thi s
I have draw n o n lecture s tha t I gave a t th e Australia n Summe r Researc h Institut e hel d
at th e Universit y o f Sydne y i n 1971an d a t th e Australia n Nationa l Universit y a t
Canberra i n 1974.
I hope tha t th e presen t accoun t wil l reall y b e o f us e t o someon e wishin g t o lear n
the subject . Th e mai n argument s ar e reasonabl y complete , thoug h a fe w ke y theorem s
are quote d withou t proof . Ther e ar e man y digressions . M y intentio n i s t o entic e th e
reader t o pursu e th e subjec t furthe r i n book s suc h a s thos e o f Swan , Roggenkam p an d
Bass ([45] , [35] , [36], [3] ) a s wel l a s i n th e researc h literature . Th e exper t wil l rec -
ognise som e ne w result s scattere d throug h th e notes .
A brief descriptio n o f th e content s ma y b e i n order . I n Lectur e 1 we describ e th e
group-theoretic settin g fro m whic h ou r subjec t arises . Lectur e 2 contain s a complet e
discussion o f relatio n module s ove r a field . Th e result s her e (bu t no t th e proofs ) g o
back t o a paper o f Gaschut z [l5 ] i n 1954. Th e nex t tw o lecture s buil d u p th e algebrai c
machinery neede d t o attac k integra l relatio n modules . Th e firs t o f thes e collect s to -
gether elementar y materia l fo r eas e o f referenc e later . Th e secon d contain s a proof o f
Swan's structur e theore m fo r projective modules . Thi s i s give n completel y modul o onl y
the nonsingularit y o f th e Carta n matrix . I n Lectur e 5 th e stud y o f relatio n module s be -
gins i n earnest . W e discuss projectiv e summand s an d introduc e th e notio n o f relatio n
cores an d th e presentatio n rank . Th e latte r i s studie d furthe r i n Lectur e 6 . The n i n
Lecture 7 w e tak e u p th e questio n o f th e numbe r o f th e abelianise d relations : a basi c
result o f Swa n i s prove d an d applie d t o thes e problems . I n th e penultimat e tw o lecture s
we stud y th e decompositio n propertie s o f relatio n cores . Th e las t lectur e place s re -
lation core s onc e agai n int o th e broade r contex t o f grou p theor y an d connect s ou r results
with genera l fact s abou t extensio n theory .
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