We are working over the finite field $\mathbb{F}_{q}$ of odd prime characteristic $p$ and of cardinality $q$ some power of $p$. We recall the symplectic group $Sp(4,\mathbb{F}_{q})$ as the group of transformations over $\mathbb{F}_{q}^{4}$ preserving a non degenerate alternate bilinear form, and we denote by $PSp(4,\mathbb{F}_{q})$ the quotient by its center. I would like to know all the possible maximal subgroups of this projective group or the initial one. A good reference may be of precious help.

  • 2
    $\begingroup$ The following paper by Aschbacher is probably the canonical reference: Aschbacher, Michael. "On the maximal subgroups of the finite classical groups." Inventiones mathematicae 76.3 (1984): 469-514. There's also a book by Kleidman and Liebeck which gives more details. In fact, §3.5 of that book gives a list of all the subgroups as far as I can tell. $\endgroup$ – Jay Taylor Feb 14 '15 at 12:26

You can find tables of the maximal subgroups of all (almost) simple classical groups in dimensions up to $12$ in the book:

The Maximal Subgroups of the Low-Dimensional Finite Classical Groups, John N. Bray,Derek F. Holt, Colva M. Roney-Dougal, Cambridge University Press, 2013.

The maximal subgroups of ${\rm Sp}_4(q)$ for odd $q$ were first classified in:

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

Here is a complete list. The notation is similar to that used in the ATLAS. There is one conjugacy class of each type except where otherwise stated.

$q^{1+2}\!:\!((q-1) \times {\rm Sp}_2(q))$ (reducible)

$q^3\!:\!{\rm GL}_2(q)$ (reducible)

${\rm Sp}_2(q)^2\!:\!2$ (imprimitive)

${\rm GL}_2(q).2$ (imprimitive)

${\rm Sp}_2(q^2)\!:\!2$ (semilinear)

${\rm GU}_2(q).2$ (semilinear)

${\rm Sp}_4(q_0).(2,r)$, $(2,r)$ classes, $q=q_0^r$, $r$ prime (subfield)

$2^{1+4}.S_5$, $2$ classes, $q$ prime, $q \equiv \pm 1 \bmod 8$

$2^{1+4}.A_5$ $q$ prime, $q \equiv \pm 3 \bmod 8$

$2.A_6$, $q$ prime, $q \equiv \pm 5 \bmod 12$, $q \ne 7$

$2.S_6$, $2$ classes, $q$ prime, $q \equiv \pm 1 \bmod 12$

$2.A_7$, $q=7$

${\rm SL}_2(q)$, $p \ge 5$, $q>7$.

  • $\begingroup$ Thanks for your answer. Mitchell gave another classification which is more geometrical and I would like to know the correspendance between the one you gave, especially the case where the maximal subgroup stabilizes a quadric, does he stabilze a subspace for example or does a subgroup of index $2$? $\endgroup$ – Silam Apr 1 '15 at 10:33

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.