广义超特殊p-群的自同构群Ⅲ

确定了广义超特殊p-群G的自同构群的结构.设|G|=p~(2n+m),|■G|=p~m,其中n≥1,m≥2,Aut_fG是AutG中平凡地作用在Frat G上的元素形成的正规子群,则(1)当G的幂指数是p~m时,(i)如果p是奇素数,那么AutG/AutfG≌Z_((p-1)p~(m-2)),并且AutfG/InnG≌Sp(2n,p)×Zp.(ii)如果p=2,那么AutG=Aut_fG(若m=2)或者AutG/AutfG≌Z_(2~(m-3))×Z_2(若m≥3),并且AutfG/InnG≌Sp(2n,2)×Z_2.(2)当G的幂指数是p~(m+1)时,(i)如果p是奇素数,那么AutG=〈θ〉■Aut_fG,其中θ的阶是(p-1)p~(m-1),且Aut_f G/Inn G≌K■Sp(2n-2,p),其中K是p~(2n-1)阶超特殊p-群.(ii)如果p=2,那么AutG=〈θ_1,θ_2〉■Aut_fG,其中〈θ_1,θ_2〉=〈θ_1〉×〈θ_2〉≌Z_(2~(m-2))×Z_2,并且Aut_fG/Inn G≌K×Sp(2n-2,2),其中K是2~(2n-1)阶初等Abel 2-群.特别地,当n=1时...

The Automorphism Group of a Generalized Extraspecial p-group III

In this paper, the automorphism group of a generalized extraspecial $p$-group $G$ is determined, where $p$ is a prime number. Assume that $|G|=p^{2n+m}$ and $|\zeta G|=p^m$, where $n \geq 1$ and $m \geq 2$. Let ${\rm Aut}_{f}G$ be the normal subgroup of ${\rm Aut}\, G$ consisting of all elements of ${\rm Aut}\, G$ which act trivially on ${\rm Frat}\, G$. Then (1) When the exponent of $G$ is equal to $p^m$, (i) If $p$ is odd, then ${\rm Aut}\, G/{\rm Aut}_{f}G\cong\mathbb{Z}_{(p-1)p^{m-2}}$ and ${\rm Aut}_{f}G/{\rm Inn}\, G\cong{\rm Sp}(2n,p)\times\mathbb{Z}_p$. (ii) If $p=2$, then ${\rm Aut}\, G={\rm Aut}_{f}G$ (when $m=2$) or ${\rm Aut}\, G/{\rm Aut}_{f}G\cong\mathbb{Z}_{2^{m-3}}\times\mathbb{Z}_2$ (when $m\geq 3$), and ${\rm Aut}_{f}G/{\rm Inn}\, G\cong{\rm Sp}(2n,2)\times\mathbb{Z}_2$. (2) When the exponent of $G$ is equal to $p^{m+1}$, (i) If $p$ is odd, then ${\rm Aut}\, G=\langle\theta\rangle\ltimes{\rm Aut}_{f}G$, where $\theta$ is of order $(p-1)p^{m-1}$, and ${\rm Aut}_{f}G/{\rm Inn}\, G\cong K\rtimes{\rm Sp}(2n-2,p)$, where $K$ is an extraspecial $p$-group of order $p^{2n-1}$. (ii) If $p=2$, then ${\rm Aut}\, G=\langle\theta_1, \theta_2\rangle\ltimes{\rm Aut}_{f}G$, where $\langle\theta_1, \theta_2\rangle=\langle\theta_1\rangle \times\langle\theta_2\rangle\cong\mathbb{Z}_{2^{m-2}}\times\mathbb{Z}_2$, and ${\rm Aut}_{f}G/{\rm Inn}\, G\cong K\rtimes{\rm Sp}(2n-2,2)$, where $K$ is an elementary abelian 2-group of order $2^{2n-1}$. In particular, ${\rm Aut}_{f}G/{\rm Inn}\, G\cong\mathbb{Z}_{p}$ when $n=1$.