# Finite group with elementary abelian centralizer

Let $G$ be group of order $q(q^{2}-1)/2$ (where $q=p^n$ is an odd prime power) such that $C_{G}(P)$ is elementary abelian for every Sylow $p$-subgroup $P$.

Is there any classifications of this type of groups? Thanks.

• Is the cyclic group of order 6 considered elementary abelian, or must $C_G(P)$ be a an elementary abelian $p$-group? – Jack Schmidt Jan 4 '14 at 7:35
• @JackSchmidt: Thanks. $C_{G}(P)$ is an elementary abelian $p$-group. – User1257 Jan 4 '14 at 7:55
• It seems reasonable to guess that these are PSL(2,q) and AΓL(1,8) for q=7. I don't see a proof, and the situation is dramatically different for q(q-1)/2. The larger order requires more coincidences; there is a plethora of such groups of order q(q-1)/2. I also wouldn't be surprised if there more exceptions besides q=7. I didn't see any good candidates though. – Jack Schmidt Jan 4 '14 at 8:22
• Note that any solvable group $G$ contains a subgroup $\Sigma$ all of whose Sylow subgroups are elementary abelian, and such that the set of prime divisors of $|\Sigma|$ is the same as the set of prime divisors of $|G|$. So, it might help you to look at Sylow $p$-subgroups of arbitrary solvable groups centralized by no more than one other Sylow $q$-subgroup. I think there may quite a few examples of these. – Alexander Gruber Jan 5 '14 at 14:36