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

## Contents

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.