I am reading a paper "A Determination of order $p^5$" by H A Bender ($p$ is an odd prime). He divides the classification in two classes, one which contains an abelian subgroup of order $p^4$ and other do not. He assumes an element of order $p$ which operates on this subgroup and this element together with a subgroup of order $p^4$ generates the group of order $p^5$. Now my problem is how to guarantee that there exists an element of order $p$ outside this subbgroup of order $p^4$.
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Hmm, does he assume $p$ odd? Otherwise this is not true (the generalized queternion groups have a unique element of order $2$, which will be contained in any subgroup of index $2$). – Tobias Kildetoft Dec 10 '13 at 11:26
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@TobiasKildetoft yes, this is mention in the question. – Steve Dec 10 '13 at 11:36
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Ahh, of course. Not sure how I missed that. – Tobias Kildetoft Dec 10 '13 at 11:37
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3I don't think it is true. For all odd primes $p$ there are nonsplit extensions of an elementary abelian group of order $p^4$ by a group of order $p$. You should check again exactly what he is claiming. – Derek Holt Dec 10 '13 at 20:33