I'm trying to anwser an exercise from Brown's book (Cohomology of groups) which states that, for $\mathbb{F}_{q}$ field with $q$ elements and $q$ a prime power, $SL_{n}(\mathbb{F}_{q})$ doesn't have periodic cohomology if $n \geq 3$ or if $q$ is not prime.
For $n \geq 3$ I've already solved. But for $SL_{2}(\mathbb{F}_{q})$ I still trying to solve. What I am trying to do is to find a non-cyclic abelian subgroup of $SL_{2}(\mathbb{F}_{q})$ which would solve the problem. Other possibility is to find a Sylow subgroup of $SL_{2}(\mathbb{F}_{q})$ which is not cyclic and not a quaternion group.
Can anyone give me a hint? Thank you
How about $G=\left\{\begin{bmatrix} 1 & x \\ 0 & 1\end{bmatrix},\, x \in \mathbb{F}_q\right\}$?
$G$ is isomorphic as a group to $(\mathbb{F}_q,+)$, so is abelian noncyclic.