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I know that there are the abelian groups $(\mathbb{Z}/p)^3$, $\mathbb{Z}/p^2 \times \mathbb{Z}/p$ and $\mathbb{Z}/p^3$, and the non-abelian Heisenberg group of matrices $ \left[ {\begin{array}{cc} 1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \end{array} } \right] $ where $a,b,c \in Z/p$.

I know that these are not all the groups in general because for $p=2$ we also have the quaternion group $H_8$ which is not in the list. Can we say anything more about the groups of order $p^3$ in general ?

Shaun
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Kieran McShane
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    For each $p$, there is a semidirect product corresponding to $\mathbb{Z} / \langle p \rangle$ acting on $\mathbb{Z} / \langle p^2 \rangle$ where the generator acts as multiplication by $1 + ap$ (where $a \in \mathbb{Z} / \langle p \rangle$ is a parameter). I'm not sure which of these are isomorphic to each other, nor am I sure whether that might actually be isomorphic to the Heisenberg group when $a \ne 0$. (For the case $a = 0$, it clearly just gives $\mathbb{Z}/\langle p^2\rangle \times \mathbb{Z}/\langle p\rangle$.) – Daniel Schepler Oct 03 '22 at 17:34
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    Please ignore my "freshman level answer" in this thread. It is the other answers you want :-) Particularly Keith Conrad's and Arturo Magidin's. – Jyrki Lahtonen Oct 03 '22 at 17:40
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    There is a complete classification of all groups of order $p^3$, see these notes by K. Conrad. See also other posts (there are many here), e.g., this one. – Dietrich Burde Oct 03 '22 at 18:11

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