# What are the maximal subgroups of GSp(2g,l)?

Is there a nice description of the maximal subgroups of $GSp_{2g}(\mathbb{F}_l)$?

When $g = 1$ this is $GL_2(\mathbb{F}_l)$, and Serre (in his 72 Inventiones paper) classifies its maximal subgroups (Borel, Normalizer of Cartans, and a few exceptional subgroups). I expect a classification for general g to be longer, but maybe its managable when g = 2?

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Just a historical comment: the classification of maximal subgroups of GL(2) for any finite field (not only Z/pZ) is due to Dickson so precedes Serre's paper by a good 70 years. –  Olivier Apr 20 '10 at 9:18

There has been a lot of systematic study in recent decades of the maximal subgroups of finite simple groups, especially groups of Lie type. This goes far beyond the special case here, but is written down in modern language. One standard reference (though no longer readily accessible outside libraries) is a monograph: MR1057341 (91g:20001) Kleidman, Peter(1-PRIN); Liebeck, Martin(4-LNDIC), The subgroup structure of the finite classical groups. London Mathematical Society Lecture Note Series, 129. Cambridge University Press, Cambridge, 1990. x+303 pp. ISBN: 0-521-35949-X 20-02 (20D06 20G40). This builds on earlier work by M. Aschbacher and others. (Kleidman left a position at Caltech to enter the financial world, but Liebeck has remained active at Imperial College, London). John Cullinan (now at Bard College), a recent Ph.D. student here in number theory, found Kleidman-Liebeck especially helpful for the special cases in his thesis. There may be some overlap between that work and what you are studying: MR2314732 (2008b:14077) 14K15 (11G10) Cullinan, John (1-BARD) Local-global properties of torsion points on three-dimensional abelian varieties. J. Algebra 311 (2007), no. 2, 736–774.

For a useful survey as of 1987: MR933365 (89c:20029) 20D05 (20E28) Seitz, Gary M. (1-OR), Representations and maximal subgroups. The Arcata Conference on Representations of Finite Groups (Arcata, Calif., 1986), 275–287, Proc. Sympos. Pure Math., 47, Part 1, Amer. Math. Soc., Providence, RI, 1987.

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"I expect a classification for general g to be longer, but maybe its managable when g = 2?"

The Experimental Mathematics paper of Dieulefait for g=2 uses this. He quotes Mitchell from 1914.

http://www.expmath.org/expmath/volumes/11/11.4/pp503_512.pdf

Kleidman and Liebeck is on Google Books. There is also a brief survey of it.

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Mitchell, H. The subgroups of the quaternary abelian linear group. Trans. Amer. Math. Soc. 15 (1914) 379-396.

You can also find complete lists of maximal subgroups of Sp(2g,l) for small l in the Atlas of Finite Groups.

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Given your tags, I'm going to suggest that you look at the papers of Larsen and Larsen and Pink. They studied the size of the image of a general compatible system of $l$-adic representations in a rather general context (following, in some sense, the paper of Serre you mention).
Given a connected semisimple algebraic group $G/\mathbb{Q}$, we can extend $G$ to a smooth group scheme over $\mathbb{Z}/[l/N]$ for $N$ sufficiently divisible. We consider the family of finite groups $G(\mathbb{F}_l)$ for $l$ >> 0. Theorem 1.1 says that for $l$ >> 0, every maximal proper subgroup of the abstract group $G(\mathbb{F}_l)$ is actually algebraic in a suitable sense. The proof depends on the classification of finite simple groups.