Automorphisms of a Class of Finite p-groups with a Cyclic Derived Subgroup

Abstract

Let p be an odd prime, and let k be a nonzero nature number. Suppose that nonabelian group G is a central extension as follows

$$1 \to G\prime \to G \to {{\mathbb{Z}}_{{p^k}}} \times \cdots \times {{\mathbb{Z}}_{{p^k}}},$$

where G′ ≅ ℤ_pk, and ζG/G′ is a direct factor of G/G′.Then G is a central product of an extraspecial p^k-group E and ζG. Let ∣E∣ = p^(2n+1)k and ∣ζG∣ = p^(m+1)k. Suppose that the exponents of E and ζG are p^k+l and p^k+r, respectively, where 0 ≤ l, r ≤ k. Let Aut_G′G be the normal subgroup of Aut G consisting of all elements of Aut G which act trivially on the derived subgroup G′, let Aut_G/ζG,ζGG be the normal subgroup of Aut G consisting of all central automorphisms of G which also act trivially on the center ζG, and let Aut_{G/ζG,ζG/G′}G be the normal subgroup of Aut G consisting of all central automorphisms of G which also act trivially on ζG/G′. Then (i) The group extension 1 → Aut_G′G → Aut G → Aut G′ → 1 is split. (ii) Aut_G′G/Aut_G/ζG,ζGG ≅ G₁ × G₂, where ${\rm{Sp}}(2n - 2,{{\bf{Z}}_{{p^k}}})\ltimes H \le {G_1} \le {\rm{Sp}}(2n,{{\bf{Z}}_{{p^k}}})$, H is an extraspecial p^k -group of order p^(2n−1)k and $({\rm{GL}}(m - 1,{{\bf{Z}}_{{p^k}}})\ltimes \mathbb{Z}_{{p^k}}^{(m - 1)})\ltimes \mathbb{Z}_{{p^k}}^{(m)} \le {G_2} \le {\rm{GL}}(m,{{\bf{Z}}_{{p^k}}})\ltimes \mathbb{Z}_{{p^k}}^{(m)}$. In particular, ${G_1} = {\rm{Sp}}(2n - 2,{{\bf{Z}}_{{p^k}}})\ltimes H$ if and only if l = k and r = 0; ${G_1} = {\rm{Sp}}(2n,{{\bf{Z}}_{{p^k}}})$ if and only if l ≤ r; ${G_2} = ({\rm{GL}}(m - 1,{{\bf{Z}}_{{p^k}}})\ltimes {\mathbb{Z}}_{{p^k}}^{(m - 1)})\ltimes {\mathbb{Z}}_{{p^k}}^{(m)}$ if and only if r = k; ${G_2} = {\rm{GL}}(m,{{\bf{Z}}_{{p^k}}})\ltimes {\mathbb{Z}}_{{p^k}}^{(m)}$ if and only if r = 0. (iii) Aut_G′G/Aut_{G/ζG,ζG/G′}G ≅ G₁ × G₃, where G₁ is defined in (ii); ${\rm{GL}}(m - 1,{{\bf{Z}}_{{p^k}}})\ltimes {\mathbb{Z}}_{{p^k}}^{(m - 1)} \le {G_3} \le {\rm{GL}}(m,{{\bf{Z}}_{{p^k}}})$. In particular, ${G_3}{\rm{= GL}}(m - 1,{{\bf{Z}}_{{p^k}}})\ltimes {\mathbb{Z}}_{{p^k}}^{(m - 1)}$ if and only if r = k; ${G_3} = {\rm{GL}}(m,{{\bf{Z}}_{{p^k}}})$ if and only if r = 0. (iv) ${\rm{Au}}{{\rm{t}}_{G/\zeta G,\zeta G/G\prime}}G \cong {\rm{Au}}{{\rm{t}}_{G/\zeta G,\zeta G}}G\rtimes {\mathbb{Z}}_{{p^k}}^{(m)}$. If m = 0, then ${\rm{Au}}{{\rm{t}}_{G/\zeta G,\zeta G}}G = {\rm{Inn}}\,G \cong \mathbb{Z}_{{p^k}}^{(2n)}$; If m > 0, then ${\rm{Au}}{{\rm{t}}_{G/\zeta G,\zeta G}}G \cong \mathbb{Z}_{{p^k}}^{(2nm)} \times \mathbb{Z}_{{p^{k - r}}}^{(2n)}$, and ${\rm{Au}}{{\rm{t}}_{G/\zeta G,\zeta G}}G/{\rm{Inn}}\,G \cong \mathbb{Z}_{{p^k}}^{(2n(m - 1))} \times \mathbb{Z}_{{p^{k - r}}}^{(2n)}$.