From: mareg@csv.warwick.ac.uk (Derek Holt)
Subject: Re: Invertible 2 X 2 matrices mod n
Date: 14 Nov 2001 01:56:11 -0800
Newsgroups: sci.math
Summary: Commutator subgroups of general linear groups

bill_pet@my-deja.com (Bill) wrote in message news:<36abe133.0111130455.249fc65c@posting.google.com>...
> mareg@primrose.csv.warwick.ac.uk () wrote in message news:<9so5pu$3jk$1@wisteria.csv.warwick.ac.uk>...
> > In article <1ddac121.0111081203.4241ad83@posting.google.com>,
> > 	olaveshta@my-deja.com (Ola) writes:
> > >"
> > >Let G_n be the group of invertible 2 X 2 matrices mod n.

...

> 
> Is it true that except for n=2 the commutator subgroup of G_n is the
> subgroup of matrices of determinant one ?
> 


No. When n=2 all elements of G_n have determinant 1, but G_n is dihedral of
order 6 and has commutator subgroup of order 3.

The derived group of GL(2,Z_n) is equal to SL(2,Z_n) if and only if n is odd.

Derek Holt