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I have the following question:

Let $G$ be a group. And $A,B\leq G$ two cyclic subgroups with $|A|=|B|=n$. When does an Automorphism $\alpha\in\mathrm{Aut}(G)$ exists such that $\alpha(A)=B$?

If $n=p$ is prime, does this changes the situation?

Thanks for help.

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2  
I really don't think that there is any sensible answer to that question without further information or assumptions! –  Derek Holt Aug 30 '12 at 22:15
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Take $n=p=2$ and $G$ a symmetric group. You get a very clear answer (iff the subgroups have the same cycle structure) which simplifies to "sometimes". –  Jack Schmidt Aug 31 '12 at 2:16
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If $n$ is a maximal prime power (i.e. $n=p^k$ and $pn$ does not divide $|G|$), then the answer is yes, and in fact then $\alpha\in\operatorname{Inn}(G)$ and $A$, $B$ need not be cyclic (Sylow theorem). In other cases (as given by Jack Smith) $A$ may be a subgroup of a cyclic subgroup of order $k n$, say. This is a property kept by automorphisms, so if $B$ does not have ist, there is no automorphism. –  Hagen von Eitzen Sep 1 '12 at 21:01

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