Suppose that H is a subgroup of G and $|H|=p^k$ for some prime p and integer k>1, how to determine whether H is cyclic? Here, G are some sporadic groups, say Co1, Co2, J4, etc.
closed as not a real question by Jack Schmidt, Derek Holt, O.L., Julian Kuelshammer, Lord_Farin Jun 18 '13 at 18:03
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I think with that limited amount of information you can't do any better than the trivial result $k=1$ implies $H$ cyclic. If you have more information about $G$, like for example that it is cyclic, then you can say more. To see that we can't beat this in the fully general case, note that you can take the direct product of a group $H$ of order $p^k$ with another group. The resulting group will have the specified subgroup, but depending on your choice of $H$ it may or may not be cyclic.