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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.

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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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$G$ is cyclic, for example. –  B. S. Jun 18 '13 at 17:23
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You need to be more specific and give more details if you expect a meaningful answer. –  Derek Holt Jun 18 '13 at 17:25
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If G is cyclic, then H is cyclic. If k=1, then H is cyclic. Otherwise the most simple way to determine whether the group is cyclic or not is to search an element that generates the whole group or to proof that such an element does not exist. –  mvcouwen Jun 18 '13 at 17:27
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Was just thinking about this problem and noticed an old professor's name. I don't know why this is surprising but it is, hey there professor Holt! –  James Jun 18 '13 at 17:27
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I have seen it, but now I am feeling sad about the "old" :-( (My age is the subject of an old Beatles song.) –  Derek Holt Jun 18 '13 at 20:08

1 Answer 1

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.

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