Preface

My goal in these lectures is the isomorphis m theor y o f linear groups over integra l do -

mains as illustrated b y th e theore m

PSLn(o)^PSLni(ol)*=*n = nl an d o = o

x

for dimension s 3 . Th e theory tha t follow s i s typical o f muc h o f th e researc h o f th e last

decade o n th e isomorphisms o f th e classica l groups ove r rings . I will start fro m scratch , as-

suming onl y basi c fact s fro m a first cours e i n algebra . I n particular, th e classica l theore m

on the simplicit y o f PSL n(F) wil l be proved (i n two differen t ways , as a matter o f fact) ,

and whateve r i s needed fro m projectiv e geometr y wil l be developed. Sinc e ou r interes t i s in

integral domains , we stay commutativ e throughout . I n reorganizin g th e literatur e fo r thes e

lectures I foun d i t possible t o exten d th e know n theor y fro m group s of linear transforma -

tions to groups of collinea r transformations , an d als o to improv e th e isomorphis m theor y

from dimension s 5 t o dimension s 3 . Thes e ne w result s are included i n what follows .

These note s evolved fro m lecture s a t th e Californi a Institut e o f Technology durin g th e

spring of 1968, from te n surve y lectures on classica l an d Chevalle y groups a t a n NSF Re-

gional Conference a t Arizon a Stat e Universit y i n March 1973, and fro m lecture s o n linea r

groups at th e Universit y o f Notre Dam e i n th e fal l o f 1973.

I would lik e t o expres s my thank s t o Olga Taussky an d Han s Zassenhaus fo r introduc -

ing me to th e linear group s many year s ago, to Warren Wong, Carl Riehm an d Ale x Hah n fo

countless discussion s o n th e subject , an d t o Ronal d Jacobowit z fo r a good conference .

NOTRE DAM E

O. T . O'MEAR A

JANUARY 1974

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