Maximal finite subgroups of Sp(2n, Q)

Maximal finite subgroups of Sp(2n, Q) Markus Kirschmer RWTH Aachen University Groups, Rings and Group-Rings July 12, 2011

Goal Characterization Goal: Classify all conjugacy classes of maximal finite symplectic matrix groups. Some tradition: All conjugacy classes of maximal finite subgroups of GL n (Q) are known up to n = 31 (for n 11: Nebe, Plesken,...) All conjugacy classes of maximal finite subgroups of GL n (D) where D is a totally definite quaternion algebra are known for n dim Q (Z(D)) 10 (Nebe)

Goal Characterization Let G < GL n (Q). F(G) = {X Q n n gxg t = X for all g G} is the form space and F >0 (G) the cone of symmetric positive definite forms. G is called symplectic if F(G) contains an invertible skewsymmetric form. If G < GL 2n (Q) is symplectic and F Q 2n 2n is invertible and skewsymmetric then there exists some x GL 2n (Q) such that F F(G x ).

Goal Characterization Let G < GL n (Q). C Q n n(g) = {X Q n n gx = Xg for all g G} commuting algebra Z(G) = {L Q 1 n L a full lattice, Lg = L for all g G} G-invariant lattices G is finite if and only if F >0 (G) and Z(G). In particular, if G is finite then {Aut(L, F ) (L, F ) Z(G) F >0 (G)} contains all maximal finite supergroups of G. Here Aut(L, F ) = {g GL n (Q) Lg = L and gfg t = F }.

Goal Characterization Let G < GL 2n (Q) be finite, symplectic and irreducible. Each F F >0 (G) induces an isomorphism C Q 2n 2n(G) F(G), c cf and F(G) Q 2n 2n sym C Q 2n 2n(G) is a totally real field. Thus, { Aut K (L, F ) (L, F ) Z(G) F >0(G), K < C Q 2n 2n(G) a minimal totally complex subfield } contains all maximal finite symplectic supergroups of G. Here Aut K (L, F ) = {g Aut(L, F ) xg = gx for all x K}.

Defnition Consequences Definition An irreducible group G < GL 2n (Q) is symplectic primitive if G is not conjugate to a subgroup of H S k for some symplectic matrix group H < GL 2n (Q). k Reasons to omit symplectic imprimitive matrix groups: Easy to construct from smaller dimensions. Usually maximal finite. Easy to recognize. has important consequences for normal subgroups...

Defnition Consequences Definition An irreducible group G < GL 2n (Q) is symplectic primitive if G is not conjugate to a subgroup of H S k for some symplectic matrix group H < GL 2n (Q). k Reasons to omit symplectic imprimitive matrix groups: Easy to construct from smaller dimensions. Usually maximal finite. Easy to recognize. has important consequences for normal subgroups...

Defnition Consequences Definition An irreducible group G < GL 2n (Q) is symplectic primitive if G is not conjugate to a subgroup of H S k for some symplectic matrix group H < GL 2n (Q). k Reasons to omit symplectic imprimitive matrix groups: Easy to construct from smaller dimensions. Usually maximal finite. Easy to recognize. has important consequences for normal subgroups...

Defnition Consequences Definition An irreducible group G < GL 2n (Q) is symplectic primitive if G is not conjugate to a subgroup of H S k for some symplectic matrix group H < GL 2n (Q). k Reasons to omit symplectic imprimitive matrix groups: Easy to construct from smaller dimensions. Usually maximal finite. Easy to recognize. has important consequences for normal subgroups...

Defnition Consequences Theorem Let G < GL 2n (Q) be symplectic primitive, irreducible and maximal finite (s.p.i.m.f. for short). 1. If N G then N Q is simple. 2. If N G and N is a p-group then (p 1) 2n. 3. Every abelian characteristic subgroup of O p (G) is cyclic. By a theorem of Philip Hall: O p (G) is a central product of G 1 and G 2 where G 1 = 2 1+2n and G 2 = 1 G 1 = 2 1+2n + and G 2 is either cyclic, (quasi-) dihedral or a generalized quaternion 2-group G 1 = p 1+2n + and G 2 is cyclic and p 2 (n 0)

Defnition Consequences Theorem Let G < GL 2n (Q) be symplectic primitive, irreducible and maximal finite (s.p.i.m.f. for short). 1. If N G then N Q is simple. 2. If N G and N is a p-group then (p 1) 2n. 3. Every abelian characteristic subgroup of O p (G) is cyclic. By a theorem of Philip Hall: O p (G) is a central product of G 1 and G 2 where G 1 = 2 1+2n and G 2 = 1 G 1 = 2 1+2n + and G 2 is either cyclic, (quasi-) dihedral or a generalized quaternion 2-group G 1 = p 1+2n + and G 2 is cyclic and p 2 (n 0)

General outline of the classification: 1. Construct the imprimitive groups from smaller dimensions. 2. Classify the s.p.i.m.f. groups: 2.1 For each possible Fitting subgroup F (G) (Hall) 2.2 For each possible layer E(G) (Atlas) 2.3 Find G as extension of F (G) = E(G), F (G) since G/F (G) < Out(F (G)). Finding G is a cohomological task. But there are some tricks to avoid cohomology...

Definition Examples Let G < GL 2n (Q) be s.p.i.m.f. and N G. We construct some hereditary order in N Q containing N as follows: 1. Set Λ 0 = N Z. 2. Let I i be the arithmetic radical of Λ i 1, i.e. I i is the intersection of all maximal right ideals of Λ i 1 that contain the discriminant of Λ i 1. 3. Let Λ i be the right order of I i in N Q. This radical idealizer process stabilizes at some hereditary order Λ (N).

Definition Examples Q 1 2n = QN V m for some irreducible QN-module V. Let L 1,..., L s represent the isomorphism classes of Λ(N) -invariant lattices in V. Further fix some F F >0 (N). Then the Generalized Bravais group B o (N) := {x N Q L ix = L i for all 1 i s and xfx t = F } satisfies N B o (N) G.

Definition Examples Example 1 Suppose N = Alt 5 and the natural character of N is a multiple of the irreducible 5-dimensional character of Alt 5, then B o (N) = C 2 Alt 6. So N cannot be a normal subgroup of some s.p.i.m.f. matrix group. Example 2 Suppose N = p 1+2n + with p 2. Then B o (N) = C 2 N.Sp 2n (p). Note that Out(N) = Sp 2n (p) C p 1. (If p is a Fermat prime then B o (N) < GL p n (p 1)(Q) is already maximal finite.)

Definition Examples Example 1 Suppose N = Alt 5 and the natural character of N is a multiple of the irreducible 5-dimensional character of Alt 5, then B o (N) = C 2 Alt 6. So N cannot be a normal subgroup of some s.p.i.m.f. matrix group. Example 2 Suppose N = p 1+2n + with p 2. Then B o (N) = C 2 N.Sp 2n (p). Note that Out(N) = Sp 2n (p) C p 1. (If p is a Fermat prime then B o (N) < GL p n (p 1)(Q) is already maximal finite.)

Some infinite families Number of conjugacy classes Some families of s.p.i.m.f. matrix groups: 1. ±PSL 2 (p) < GL p 1 (Q) if p 1 mod 4 2. SL 2 (p) < GL p+1 (Q) if p 11 and p 1 mod 4 3. QD 2 n < GL 2 n 1(Q) for n 5 4. 2 1+2n +.(O + 2n (2) : 2) < GL 2 n+1(q)

Some infinite families Number of conjugacy classes Number of conjugacy classes of maximal finite irreducible symplectic matrix groups in GL 2n (Q): 2n primitive imprimitive total 2 2 (1) 0 (1) 2 (2) 4 5 (2) 1 (1) 6 (3) 6 2 (4) 2 (2) 4 (6) 8 21 (5) 7 (4) 28 (9) 10 3 (6) 2 (2) 5 (8) 12 23 (11) 9 (8) 42 (19) 14 3 (9) 2 (3) 5 (12) 16 63 (22) 28 (9) 91 (31) 18 6 (10) 2 (7) 8 (17) 20 26 (21) 10 (10) 36 (31) 22 4 (9) 2 (3) 6 (12)