Linear representations of random groups

Linear Representations of Finite Groups

QUASI-PERMUTATION REPRESENTATIONS OF METACYCLIC 2-GROUPS

By a quasi-permutation matrix we mean a square matrix over the complex field C with non-negative integral trace. Thus, every permutation matrix over C is a quasipermutation matrix. For a given finite group G, let p(G) denote the minimal degree of a faithful permutation representation of G (or of a faithful representation of G by permutation matrices), let q(G) denote the minimal degree of a fa...

Some Representations of Special Linear Groups

Mat r x ( F q ) o f a l l r b y r m a t r i c e s o v e r F q , b y C t h e g r o u p G L t i ( F q ) o f t h e u n i t s of M, and by S the special linear subgroup SL , (F q ) o f C . F o r a n a r b i t r a r y field F containing F q , l e t U s t a n d f o r t h e ( c o m m u t a t i v e ) p o l y n o m i a l a l g e b r a F[xl, , x r ] , a n d c o n s i d e r U g r a d e d a s u s u a l : U...

Groups with No Nontrivial Linear Representations: Corrigenda

Professor B.A.F. Wehrfritz has kindly drawn my attention to some errors in the above paper [1]. Principally, there is an invalid assertion of complete irreducibility used in Theorem 2.1 and its corollary. Thus the expression " 1-dimensional f-representation" should be replaced by "Abelian 6-representation" in both statements. This amended conclusion is still adequate for subsequent results. In ...

Groups with No Nontrivial Linear Representations

Our aim is to initiate the study of groups G whose finite-dimensional linear representations G —* GLn(t) over a field I are all necessarily trivial. This note has three main features: (1) We survey the existing literature on such groups. Noteworthy here is the surprising interaction with the phenomenon of acyclic (homologically trivial) groups. For further discussion see Remark 1.10 below. (2) ...