# $SL_{2}(\mathbb{F}_{q})$ does not have periodic cohomology.

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.