From: "Charles Matthews" <charles@sabaki.demon.co.uk>
Subject: Re: Closed subgroups of GL_n(R)
Date: Mon, 22 Jan 2001 19:41:30 -0000
Newsgroups: sci.math
Summary: Closed subgroups are Lie subgroups

David Kushilevitz wrote in message ...

>What is known about the closed subgroups of GL_n(R) for n>=1 ?
>Is there some characterization of such groups ?


They are Lie subgroups (a theorem of E. Cartan).  They needn't be connected.
The connected ones are in 1-1 correspondence with a subset of the lie
subalgebras of gl(n, R).  Every finite group can occur for some n.  There
are very many infinite discrete subgroups.  I believe there is a natural
topology on the set of closed subgroups.  There might be more to say.

Charles