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Let $p$ be a prime number and $n$ an integer such that $p\geq n$. Let $P(n)$ denotes the number of partitions of $n$. Can we conclude from Theorem 1.1 and Theorem 1.3 in the reference FINITE_p-GROUPS_OF_MAXIMAL_CLASS_AND_EXPONENT_p, that the number of isomorphism classes of splits extensions of $( \mathbb{Z} / p \mathbb{Z} )^n$ by $\mathbb{Z} / p \mathbb{Z} $ with a non abelian middle groups is $P(n)-1$ ?. Does the subgroup $( \mathbb{Z} / p \mathbb{Z} )^n$ should be here characteristic in $(\mathbb{Z}/p\mathbb{Z})^{n}\rtimes \mathbb{Z}/p\mathbb{Z}$ ?.

Any help would be appreciated so much. Thank you all.

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    $\begingroup$ What's the question? $\endgroup$
    – David Handelman
    Commented Jan 21, 2020 at 23:52
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    $\begingroup$ The formula you have in mind doesn't even work for $n=1$. $P(1)=1$ so $P(1)-1=0$ but there is a split extension. Note, the subgroup might not be characteristic, for example when it's a direct product. $\endgroup$
    – verret
    Commented Jan 22, 2020 at 0:24
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    $\begingroup$ Thank you verret but I m focus here on the case $n\geq 2$ and the midle groups of the splits extensions must be non abelian. $\endgroup$
    – Nourr Mga
    Commented Jan 22, 2020 at 0:32

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If I understand correctly, you are counting the conjugacy classes of elements of order $p$ in $\operatorname{GL}(n,\mathbb{F}_p) $ — or equivalently, of matrices $N$ in $\operatorname{M}_n(\mathbb{F}_p) $ with $N^p=0$, but $N\neq 0$. If $p\geq n$ you get all nilpotent nonzero matrices, hence indeed $P(n)-1$ conjugacy classes using Jordan normal form, but this is false for $p<n$: for $p=2$, for instance, you get $[n/2]$ (corresponding to partitions with only 1 and 2, and at least one 2).

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    $\begingroup$ Also, in the case when a subgroup $C_p^{n-1}$ is centralized by the $C_p$, the subgroup $C_p^n$ is not characteristic. $\endgroup$
    – Derek Holt
    Commented Jan 22, 2020 at 9:03
  • $\begingroup$ Thanks for noticing. What do you think of my answer?. Is there an error in one of the answer stages? $\endgroup$
    – Nourr Mga
    Commented Jan 22, 2020 at 23:46

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