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

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.

Over the integers
For any $$m > 0$$ the group $$H_m(SL(2,q);\mathbb{Z})$$ depends only on the congruence class of $$m$$ mod 4. Note that $$H_0(SL(2,q);\mathbb{Z}) = \mathbb{Z}$$ is anomalous.

$$H_3(SL(2,q);\mathbb{Z})$$ is the trickiest to describe. Here are the rules:


 * For odd $$q$$, it is a finite cyclic group whose order is the $$\{2,3,5\}$$-part of the order of $$SL(2,q)$$, i.e., the product of the largest powers of 2, 3, and 5 dividing $$|SL(2,q)| =q^3 - q$$.
 * For $$q = 2$$, it is $$\mathbb{Z}/6\mathbb{Z}$$.
 * For $$q = 4$$, it is $$\mathbb{Z}/30\mathbb{Z}$$.
 * For $$q = 8$$, it is $$\mathbb{Z}/126\mathbb{Z}$$.
 * For $$q = 2^r, r \ge 4$$, it is the finite cyclic group whose order is the odd part of $$q^3 - q$$, i.e., the unique largest odd divisor of $$q^3 - q$$.