广义超特殊p-群的自同构群(Ⅱ)

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

The Automorphism Group of a Generalized Extraspecial p-Group(Ⅱ)

In this paper,the automorphism group of a generalized extraspecial p-groupG is determined again,where p is a prime number.Assume that |G| = p^2n+mand |ζG| = p^m,where n≥1 and m≥2.Let Aut_cG be the normal subgroup of AutGconsisting of all elements of AutG which act trivially onζG.Then（ⅰ） If p is odd,then AutG =〈θ〉Aut_cG,whereθis of order（p - 1）p^m-1;Ifp = 2,then AutG =〈θ₁,θ₂〉 Aut_cG,where〈θ₁,θ₂〉=〈θ₁〉×〈θ₂〉≌Z_2m-2×Z₂.（ⅱ） If the exponent of G is equal to p^m,then Aut_cG/InnG≌Sp（2n,p）.（ⅲ） If the exponent of G is equal to p^m+1,then Aut_cG/InnG≌K Sp（2n - 2,p）,where K is an extraspecial p-group of order p^2n-1（If p is odd） or an elementary abelian2-group of order 2^2n-1.In particular,Aut_cG/InnG≌Z_p when n=1.