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