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]