关于一类中心循环的有限P-群的自同构群

确定了一类中心循环的有限p-群G的自同构群.设G=X_3(p~m)~(*n)*Z_(p~(m+r)),其中m≥1,n≥1和r≥0,并且X_3(p~m)=x,y|x~(p~m)=y~(p~m)=1,x,y]~(p~m)=1,x,x,y]]=y,x,y]]=1.Aut_nG表示Aut G中平凡地作用在N上的元素形成的正规子群,其中G'≤N≤ζG,|N|=p~(m+s),0≤s≤r,则(i)如果p是一个奇素数,那么AutG/Aut_nG≌Z_(p~((m+s-1)(p-1))),Aut_nG/InnG≌Sp(2n,Z_(p~m))×Z_(p~(r-s)).(ii)如果p=2,那么AutG/Aut_nG≌H,其中H=1(当m+s=1时)或者Z_(2~(m+s-2))×Z_2(当m+s≥2时).进一步地,Aut_nG/InnG≌K×L,其中K=Sp(2n,Z_(2~m))(当r0时)或者O(2n,Z_(2~m))(当r=0时),L=Z_(2~(r-1))×Z_2(当m=1,s=0,r≥1时)或者Z_(2~(r-s)).

On the Automorphism Group of a Class of Finite p-Groups with Cyclic Center

The automorphism group of a class of finite $p$-groups with cyclic center is determined. Let $G=X_{3}(p^m)^{*n}*\mathbb{Z}_{p^{m+r}}$, where $m\geq 1,\ n\geq 1,\ r\geq 0$, and $$ X_3(p^m)=\langle x,y \mid x^{p^m}=y^{p^m}=1,\ x,y]^{p^m}=1,\ x,x,y]]=y,x,y]]=1\rangle. $$ Let $\mathrm{Aut}_n G$ be the normal subgroup of $\mathrm{Aut}\,G$ consisting of all elements of $\mathrm{Aut}\,G$ which act trivially on $N$, where $G''\leq N \leq \zeta G$ and $|N|=p^{m+s},\ 0\leq s \leq r$. Then (i) If $p$ is odd, then $\mathrm{Aut}\,G/\mathrm{Aut}_nG\cong \mathbb{Z}_{p^{m+s-1}(p-1)}$ and $\mathrm{Aut}_nG/\mathrm{Inn}\,G\cong \mathrm{Sp}(2n,\mathbb{Z}_{p^m})\times \mathbb{Z}_{p^{r-s}}$. (ii) If $p=2$, then $\mathrm{Aut}\,G/\mathrm{Aut}_nG\cong H$, where $H=1$ (if $m+s=1$) or $\mathbb{Z}_{2^{m+s-2}}\times\mathbb{Z}_2$ (if $m+s\geq 2$). Furthermore, $\mathrm{Aut}_nG/\mathrm{Inn}\,G\cong K\times L$, where $K=\mathrm{Sp}(2n,\mathbb{Z}_{2^m})$ (if $r>0$) or $\mathrm{O}(2n,\mathbb{Z}_{2^m})$ (if $r=0$), $L=\mathbb{Z}_{2^{r-1}}\times \mathbb{Z}_2$ (if $m=1,\ s=0,\ r\geq 1$) or $\mathbb{Z}_{2^{r-s}}$.