The Automorphism Group of a Class of Nilpotent Groups with Infinite Cyclic Derived Subgroups

Abstract

The automorphism group of a class of nilpotent groups with infinite cyclic derived subgroups is determined. Let G be the direct product of a generalized extraspecial ℤ-group E and a free abelian group A with rank m, where

$$E = \left\{ {\left( {\begin{array}{*{20}{c}} 1&{k{\alpha _1}}&{k{\alpha _2}}& \cdots &{k{\alpha _n}}&{{\alpha _{n + 1}}} \\ 0&1&0& \cdots &0&{{\alpha _{n + 2}}} \\ \vdots & \vdots & \vdots &{}& \vdots & \vdots \\ 0&0&0& \cdots &1&{{\alpha _{2n + 1}}} \\ 0&0&0& \cdots &0&1 \end{array}} \right)\left| {{\alpha _i} \in Z,i = 1,2,...,2n + 1} \right.} \right\},$$

where k is a positive integer. Let Aut_G′G be the normal subgroup of AutG consisting of all elements of AutG which act trivially on the derived subgroup G′ of G, and Aut _G/ζG,ζG G be the normal subgroup of AutG consisting of all central automorphisms of G which also act trivially on the center ζG of G. Then (i) The extension 1 → Aut_G′G → AutG → AutG′ → 1 is split. (ii) Aut_G′G/Aut _G/ζG,ζG G ≅ Sp(2n, Z) × (GL(m, Z) × (ℤ)^m). (iii) Aut _G/ζG,ζG G/InnG ≅ (ℤ _k )²ⁿ ⊕ (ℤ _k )^2nm.