From: Jim Heckman
Subject: Re: Q: Simple groups
Date: Tue, 30 Nov 1999 21:17:33 GMT
Newsgroups: sci.math.research
Keywords: simplicity of linear groups
In article <3843A162.7FCF359D@physik.uni-osnabrueck.de>,
Roland Franzius wrote:
>
> hi everybody,
> who can tell me if Sl(2,F_5) is a simple group?
Do you mean the special (determinant=1) linear group of the
2-dimensional vector space over the finite field of order 5? (Common
ways to write this include SL(2,5), SL_2(5), and variations, such as
yours, with small 'l' instead of capital 'L' and/or 'F_5' instead of '5'.)
If so, the answer is "no". However, its factor group by its center
(Z={diag(1,1), diag(4,4)}) -- PSL(2,5)=SL(2,5)/Z -- is indeed a simple
group, isomorphic to the alternating group on 5 letters, Alt(5) = A_5.
The groups PSL(n,q), the finite Projective Special Linear groups -- often
abbreviated just L(n,q) -- are all simple except for L(2,2) = Sym(3) = S_3
and L(2,3) = Alt(4) = A_4.
IIRC, the only finite Projective Special Linear groups that are isomorphic
to alternating groups are: L(2,4) = L(2,5) = Alt(5) = A_5; L(2,9) = Alt(6) =
A_6; and L(4,2) = Alt(8) = A_8.
--
~~ Jim Heckman ~~
-- "As I understand it, your actions have ensured that you will never
see Daniel again." -- Larissa, a witch-woman of the Lowlands.
-- "*Everything* is mutable." -- Destruction of the Endless
Sent via Deja.com http://www.deja.com/
Before you buy.
==============================================================================
[Remark: PSL(n,F) is simple for all fields, finite or not, except for the
two cases PSL(2,F_2) and PSL(2, F_3). There is a nice discussion of the
geometry of linear groups in Jacobson's "Basic Algebra I", Freeman (1974)
--djr]