Extraspecial p-groups

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A $p$-group $G$ is called extraspecial if its center $Z$ is cyclic of order $p$, and the quotient $G/Z$ is a non-trivial elementary abelian $p$-group. Extraspecial groups of order $p^{1+2n}$ are often denoted by the symbol $p^{1+2n}$.

Remark. Since $G/Z$ is elementary abelian, we have that $Z⩾G^{\prime }$ and $Z⩾\Phi (G)$. If $G$ is abelian, then $G\cong {\mathbb{Z}}_p$, which is impossible. So $G$ is non-abelian and $G^{\prime },\Phi (G)$ are non-trivial. Therefore, it deduces that $\mathrm{Z}(G)=G^{\prime }=\Phi (G)\cong {\mathbb{Z}}_p$ and so $G$ is a special p-group.

If $n$ is a positive integer there are two extraspecial groups of order $p^{1+2n}$, which for $p$ odd are given by

• The central product of $n$ extraspecial groups of order $p^3$, all of exponent $p$. This extraspecial group has exponent $p$, denoted by $p_{+}^{1+2m}$.
  • It is known that $p_{+}^{1+2}\cong {\mathrm{SL}}_3(p{)}_p$, where ${\mathrm{SL}}_3(p{)}_p$ is a Sylow $p$-subgroup of ${\mathrm{SL}}_3(p)$. Thus $p_{+}^{1+2m}$ is the central product of $m$ copies of $\mathrm{SL}(3,p{)}_p$ for odd prime $p$, see here.
  • On the other hand, $p_{+}^{1+2}=⟨a,b|a^p=b^p=c^p=1,[a,b]=c⟩$, see here. With this definition, it is easy to show that $\mathrm{Aut}(p_{+}^{1+2})\cong {\mathrm{GL}}_2(p)$.
  • $\mathrm{Out}(p_{+}^{1+2m})\cong {\mathrm{CSp}}_{2m}(p)\cong {\mathrm{Sp}}_{2m}(p):{\mathbb{Z}}_{p-1}$. See here.
• The central product of n extraspecial groups of order $p^3$, at least one of exponent $p^2$. This extraspecial group has exponent $p^2$, denoted by $p_{-}^{1+2m}$.

When $p=2$,

• there are two extraspecial groups of order $8=2^3$, which are given by:
  • The dihedral group $D_8$, which has $2$ elements of order $4$.
  • The quaternion group $Q_8$, which has $6$ elements of order $4$.
• If $n$ is a positive integer there are two extraspecial groups of order $2^{1+2n}$, which are given by
  • The central product of $n$ extraspecial groups of order $8$, an odd number of which are quaternion groups.
  • The central product of $n$ extraspecial groups of order $8$, an even number of which are quaternion groups.