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

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 ={(1 kα1 kα2 ··· kα_nαn+1 0 1 0 ··· 0 αn+2...............000...1 α2n+1000...01|αi∈ Z, i = 1, 2,..., 2 n + 1},where k is a positive integer. 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 of G, and AutG/ζ G,ζ GG be the normal subgroup of Aut G consisting of all central automorphisms of G which also act trivially on the center ζ G of G. Then（i） The extension 1→ AutG' G→ AutG→ Aut（G'）→ 1 is split.（ii） AutG' G/AutG/ζ G,ζ GG≌Sp（2 n, Z） ×(GL（m, Z）■（Z）m).（iii） AutG/ζ G,ζ GG/Inn G≌（Zk）2n⊕（Z）2nm.