Group cohomology of special linear group of degree two over a finite field

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This article gives specific information, namely, group cohomology, about a family of groups, namely: special linear group of degree two. This article restricts attention to the case where the underlying ring is a finite field.
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This article describes the group cohomology of the special linear group of degree two over a finite field. The order (size) of the field is q, and the characteristic prime is p. q is a power of p. We denote the group as SL(2,q) or SL_2(q).

Each SL(2,q) is a finite group with periodic cohomology, and the period is 4. We can confirm this from the subgroup structure of special linear group of degree two over a finite field: all the Sylow subgroups are either cyclic groups or generalized quaternion groups.

The cases q = 2 (see group cohomology of symmetric group:S3) and q = 3 (see group cohomology of special linear group:SL(2,3)) are somewhat anomalous.

Particular cases

q (field size) p (underlying prime, field characteristic) Case for q Group SL(2,q) Order of the group (q^3 - q) Group cohomology page
2 2 even symmetric group:S3 6 group cohomology of symmetric group:S3
3 3 odd special linear group:SL(2,3) 24 group cohomology of special linear group:SL(2,3)
4 2 even alternating group:A5 60 group cohomology of alternating group:A5
5 5 odd special linear group:SL(2,5) 120 group cohomology of special linear group:SL(2,5)
7 7 odd special linear group:SL(2,7) 336 group cohomology of special linear group:SL(2,7)
8 2 even projective special linear group:PSL(2,8) 504 group cohomology of projective special linear group:PSL(2,8)
9 3 odd special linear group:SL(2,9) 720 group cohomology of special linear group:SL(2,9)

Homology groups for trivial group action

FACTS TO CHECK AGAINST (homology group for trivial group action):
First homology group: first homology group for trivial group action equals tensor product with abelianization
Second homology group: formula for second homology group for trivial group action in terms of Schur multiplier and abelianization|Hopf's formula for Schur multiplier
General: universal coefficients theorem for group homology|homology group for trivial group action commutes with direct product in second coordinate|Kunneth formula for group homology

Over the integers

For any m > 0 the group H_m(SL(2,q);\mathbb{Z}) form a periodic sequence, i.e., the group is a finite group with periodic cohomology. Note that H_0(SL(2,q);\mathbb{Z}) = \mathbb{Z} is anomalous.

Congruence class of m mod 4 H_m(SL(2,q);\mathbb{Z}) for q = 2 H_m(SL(2,q);\mathbb{Z}) for q = 3
1 \mathbb{Z}/2\mathbb{Z} \mathbb{Z}/3\mathbb{Z}
2 0 0
3 \mathbb{Z}/6\mathbb{Z} \mathbb{Z}/24\mathbb{Z}
0 0 0

For q > 3, H_1(SL(2,q); \mathbb{Z}) = H_2(SL(2,q); \mathbb{Z}) = 0. H_3 exhibits interesting behavior.