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If a prime power divides order of a group, then the group contains a subgroup of that order. We can consider the following natural question: if a prime power divides order of a group, does there exists a subgroup of that index in the group? The answer is NO, as $|A_5|=2^2.3.5$ and $A_5$ has no subgroup of index $3$.

Further, $A_6$ is also an interesting example in the sense that for any prime divisor of its order, $A_6$ has no subgroup of that index.

My question is: is there a finite group $G$ such that for any prime power divisor of $|G|$, $G$ has no subgroup of that index?

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You already gave such an example. $A_6$ has no subgroups of prime power index - see this page for a list of all subgroups of $A_6$.

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  • $\begingroup$ @Thanks for noticing it. $\endgroup$ – Beginner Feb 3 '14 at 8:21

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