Let $G$ be a finite group which has not any cyclic subgroups of order $p^2$, for each prime dividing $\vert G\vert$. What is the most that can be said about the structure of $G$?
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2$\begingroup$ Not really research level, the condition is too restrictive. Such a group G is either cyclic or metacyclic, of squarefree order. $\endgroup$– Geoff RobinsonCommented Sep 8, 2016 at 11:35
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$\begingroup$ I changed the conditions to a more general case. Maybe it is easier to think about it. $\endgroup$– H.ShahsavariCommented Sep 8, 2016 at 11:58
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$\begingroup$ Sylow 2's are elementary abelian so Walter's theorem applies and gives some examples with simple $G$ -- for instance $PSL(2,q)$ for certain values of $q$ and $J_1$. $\endgroup$– Nick GillCommented Sep 8, 2016 at 12:22
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1$\begingroup$ My comment above was about the original question. $\endgroup$– Geoff RobinsonCommented Sep 8, 2016 at 13:57
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$\begingroup$ $G$ satisfies the above condition iff any its Sylow subgroup is cyclic or generalized quaternion. Such $G$ are described in know papers of H. Zassenhaus and M. Suzuki. There are among them nonsolvable groups, for example, SL(2,5). $\endgroup$– yakovCommented Sep 16, 2016 at 8:57
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1 Answer
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Maybe the best way to approach this question is to study $F^*(G)$, the generalized Fitting subgroup of $G$. This splits into two parts:
$F(G)$ -- the Fitting subgroup. This is a direct product of $p$-groups, and your condition requires that they all have exponent $p$. The theory of such $p$-groups is extensive and there are plenty of examples: elementary abelian groups, extraspecial groups, etc.
$E(G)$ -- the Layer. This is a direct product of quasisimples. As I said in the comment above, your condition implies that the Sylow $2$-subgroups are elementary abelian so each of these quasisimples $T$ will have $T/Z(T)$ isomorphic to one of the groups listed in Walter's theorem. These are $PSL_2(q)$ with $q\equiv 3,5\pmod 8$, or else with $q$ even; $J_1$ (one of Janko's sporadic groups); ${^2G_2(3^{2a+1})}$ (one of the families of Ree groups). Your condition is stronger than Walter's, so some of these can be excluded. (For instance the Ree groups have elements of order $9$ and so can be excluded; similarly, in the $PSL(2,q)$ case, your condition requires that $q(q^2-1)$ is not divisible by an odd square.) But, anyway, you could check these.
So there are plenty of possibilities for $F^*(G)$ and these should be classifiable (at least up to the classification of $p$-groups of exponent $p$). Now Bender's famous theorem about $F^*(G)$ -- that $C_G(F^*(G)) = Z(F^*(G))$ -- tells you the possibilities for the group $G$ itself (at least in theory).
To obtain a complete answer using Bender's theorem, you would need to know the structure of the automorphism group of all parts of $F^*(G)$ -- this is fine for the quasisimples involved in the layer, but more tricky for the $p$-groups part. You'd also need to consider the original question for subgroups of the symmetric group (to deal with the situation where the layer contained multiple copies of the same quasisimple).
