From: mareg@mimosa.csv.warwick.ac.uk () Subject: Re: Images of GL_2(C) Date: Sun, 22 Jul 2001 16:59:45 +0000 (UTC) Newsgroups: sci.math Summary: Composition factors of general linear groups In article , sharonsela@hotmail.com (Sharon) writes: >rusin@vesuvius.math.niu.edu (Dave Rusin) wrote in message news:<9jckou$fl$1@news.math.niu.edu>... >> In article , >> Sharon wrote: >> >Is there some characterization about which finite groups are >> >homomorphic images of the group GL_2(C) ? >> >> Hint: If x is in GL_2(C) and N is an integer, you can find a y >> with y^N = x . > >Ok thanks I got it , GL_2(C) is divisible so it has only trivial finite >image. >Let me ask the same question also about GL_2(R) . For any n>=2 and any field K, except in the two cases n=2, |K|=2, and n=2, |K|=3, the group SL(n,K) is perfect, and PSL(n,K) is simple. So, for any infinite field K, and n>=2, SL(n,K) has no proper normal subgroups of finite index. (In fact that is also true for n=1.) Hence any normal subgroup of GL(n,K) of finite index must contain SL(n,K). So the finite homomorphic images of GL(n,K) (K infinite) are the same as those of K#, the multiplicative group of K-{0}. That should make it easier! Derek Holt.