From: George McNinch Subject: Re: Algebraic Groups Date: 28 Dec 1999 16:45:42 -0500 Newsgroups: sci.math.research Keywords: algebraic groups resemble linear groups and finite groups >>>>> "Victor" == Victor S Miller writes: Victor> Can someone point me to a proof of the theorem that a Victor> group variety (or probably a group scheme, maybe subject Victor> to some reasonable hypotheses to rule out pathological Victor> cases) can be formed by extensions of finite groups, Victor> abelian varieties and matrix groups (something which Victor> becomes isomorphic to the group of n by n matrices after Victor> possibly a finite extension of the defining field)? If the group variety is affine, such a group is an extension of a connected linear group by a finite group. Here, a linear group just means a closed subgroup of GL_n (n by n non-singular matrices). This amounts to the statement that the connected component of the identity has finite index in G (since G is quasicompact), and that an affine group variety has a faithful matrix representation. For these facts, you can see e.g. A. Borel, Linear Algebraic Groups, Springer-Verlag GTM 126. But the connected part needn't become isomorphic to some GL_n after a field extension, or even be built up by extensions of such groups (e.g. the "additive group" will not, nor will various classical groups like the symplectic group Sp_2n, ...) One can say more about the connected component group, though. Namely, it has a maximal normal, connected, solvable subgroup for which the quotient group is more-or-less a product of various (almost) simple groups (like Sp_2n, SL_n, SO_n, ...) For this, see again Borel's book. When G is no longer affine, I think that J.-P. Serre, "Algebraic Groups and Class Fields" may contain the statement you want (I could be mistaken as I don't have it handy right now). But I think at least Serre should have there a reference for what you want. -- George McNinch | __O www.nd.edu/~gmcninch | _-\<,_ Dept. Math, Univ Notre Dame | (_)/ (_) |-------------- ============================================================================== From: David Joyner Subject: Re: Algebraic Groups Date: Tue, 28 Dec 1999 15:24:58 -0500 Newsgroups: sci.math.research "Victor S. Miller" wrote: [original post quoted --djr] Chevalley's Theorem, Ch III, section 3.3, in Shaferavich, B.A.G., 1977 ============================================================================== From: Axel Schmitz-Tewes Subject: Re: Algebraic Groups Date: Wed, 29 Dec 1999 10:01:50 +0100 Newsgroups: sci.math.research George McNinch schrieb: > .... > > When G is no longer affine, I think that J.-P. Serre, "Algebraic > Groups and Class Fields" may contain the statement you want (I could > be mistaken as I don't have it handy right now). If you have a group variety G, you can look at the co-group \Gamma of global sections of the structure sheaf. From that you get a closed immersion Spec(\Gamma) -> G of group varieties which imbeds Spec(\Gamma) as a normal subgroup. The quotient (for the existence there is something to prove!) should be an abelian variety and Spec(\Gamma) is an affine variety for which you can use the results posted before. regards, axel