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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

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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.

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  • $\begingroup$ You are right, I edited. $\endgroup$
    – Mindlack
    Jan 22 '20 at 13:34
  • $\begingroup$ Oh, that's right, thanks. I had considered this group, but I dont know why, I assumed this was cyclic. $\endgroup$ Jan 22 '20 at 13:46

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