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Does someone have an idea about a possible way to count the number of normal subgroup that a group of order $p ^n $ has ($n \in \mathbb{N}$ )? Is there anyway we can count the maximal subgroups it has (i.e.- the groups of order $p^{n-1} $ ? ) ?

Thanks in advance!

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Counting maximal subgroups is equivalent to counting p-cycles in the elementary abelian group $P/\phi(P)$. –  peoplepower Oct 1 '12 at 23:38
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It's going to depend a lot on the group, isn't it? Just among groups of order 8, the answer varies from 4 to 16. –  Gerry Myerson Oct 1 '12 at 23:56
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In case you aren't the same person, there is a thread about this on MO. mathoverflow.net/questions/108581/108584#108584 –  Alexander Gruber Oct 2 '12 at 4:09

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