# Group cohomology of special linear group:SL(2,5)

## Contents

View group cohomology of particular groups | View other specific information about special linear group:SL(2,5)

## Family contexts

Family name Parameter values General discussion of group cohomology of family
double cover of alternating group degree $n = 5$, i.e., the group $2 \cdot A_5$ group cohomology of double cover of alternating group
special linear group of degree two over a finite field of size $q$ $q = 5$, i.e., field:F5, i.e., the group is $SL(2,5)$ group cohomology of special linear group of degree two over a finite field

## 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

The homology groups over the integers are as follows:

$H_m(SL(2,5);\mathbb{Z}) = \left \lbrace \begin{array}{rl} \mathbb{Z}, & m = 0 \\ \mathbb{Z}/120\mathbb{Z}, & m \equiv 3 \pmod 4 \\ 0, & m \equiv 0,1,2 \pmod 4, m > 0 \\\end{array}\right.$

The sequence of homology groups for positive degrees has a period of 4, which is in keeping with the fact that $SL(2,5)$ is a finite group with periodic cohomology, as are all special linear groups of degree two over a finite field.