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

This article gives specific information, namely, group cohomology, about a particular group, namely: special linear group:SL(2,5).
View group cohomology of particular groups | View other specific information about special linear group:SL(2,5)

This article describes the group cohomology of 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.