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Are there any relatively common notations for the two non-isomorphic nonabelian groups of order $p^3$ where $p$ is a prime number? I remember reading some notations like $p_+^{1+2}$ and $p_-^{1+2}$. Where can I find the definition of these notations?

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2 Answers 2

up vote 3 down vote accepted

These groups are the so-called extra-special groups of order $p^{3}.$ For each prime $p$ there are exactly two isomorphism types of such groups. The case $p=2$ has been discussed in another answer. When $p$ is odd, the two isomorphism types are distinguished by the fact that one has exponent $p,$ the other has exponent $p^{2}.$ I believe an explanation of the notation you refer to may be found in the "Atlas of Finite Groups".

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Do you know $2_+^{1+2}$ denotes $D_8$ or $Q_8$? –  Binzhou Xia Aug 8 '12 at 13:59
    
I don't know I'm afraid! –  Geoff Robinson Aug 8 '12 at 14:07
3  
+ has exponent $p$ when $p$ is odd, and is dihedral when $p=2$. This is the atlas page xx, right column. (Even number of quaternionic factors includes 0 such factors). This is the same convention used by GAP. –  Jack Schmidt Aug 8 '12 at 15:46
    
Thanks so much! –  Binzhou Xia Aug 8 '12 at 16:41

The nonabelian groups of order $8$ are $Q_8$ and $D_8$. The nonabelian groups of order $p^3$ for $p>2$ prime are $H_3(\mathbb{Z}_p)$ and $G_p$, where $H_3(\mathbb{Z}_p)$ is the Heisenberg group consisting of upper unitriangular matrices with entries in $\mathbb{Z}_p$ and

$$G_p = \left\{ \begin{pmatrix} a & b \\ 0 & 1 \end{pmatrix}\, :\, a,b \in \mathbb{Z}/p^2\mathbb{Z},\ a \equiv 1 \pmod p \right\}$$

For proofs, see here.

I've never come across the notations you mention, which isn't to say they don't exist.

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