Group cohomology of special linear group:SL(2,3)

This article describes the group cohomology of special linear group:SL(2,3).

Over the integers
The homology groups over the integers are as follows:

$$H_m(SL(2,3);\mathbb{Z}) = \left \lbrace \begin{array}{rl} \mathbb{Z}, & m = 0 \\ \mathbb{Z}/3\mathbb{Z}, & m \equiv 1 \pmod 4 \\ \mathbb{Z}/24\mathbb{Z}, & m \equiv 3 \pmod 4 \\ 0, & m \mbox{ even}, m > 0 \\\end{array}\right.$$

The homology groups have a period of 4, which is in keeping with the fact that $$SL(2,3)$$ is a finite group with periodic cohomology.

Computation of homology groups
The homology groups for trivial group action on the integers can be computed using the Hap package (if the package is installed but not automatically loaded, load it using LoadPackage("hap");), specifically its GroupHomology function. The function returns a list of numbers which are the orders of cyclic groups whose external direct product is the desired homology group.

First homology group
The first homology group, which is also the abelianization, can be computed as follows:

gap> GroupHomology(SL(2,3),1); [ 3 ]

This says that $$H_1(SL(2,3);\mathbb{Z}) = \mathbb{Z}/3\mathbb{Z}$$.

Second homology group
The second homology group, which is also the Schur multiplier, can be computed as follows:

gap> GroupHomology(SL(2,3),2); [ ]

This says that $$H_2(SL(2,3);\mathbb{Z}) = 0$$.

First few homology groups
gap> List([1..8],i -> [i,GroupHomology(SL(2,3),i)]); [ [ 1, [ 3 ] ], [ 2, [ ] ], [ 3, [ 8, 3 ] ], [ 4, [  ] ], [ 5, [ 3 ] ],  [ 6, [  ] ], [ 7, [ 8, 3 ] ], [ 8, [  ] ] ]