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

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.