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 .-group E and a free abelian group A with rank m, where E=10.00k.11.00k.20.00....k.n0.10.n+1.n+2..2n+11 .i.Z,i=1,2,...,2n+1, where k is a positive integer. Let AutG.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/.G,.G G be the normal subgroup of AutG consisting of all central automorphisms of G which also act trivially on the center .G of G. Then (i) The extension 1 . AutG.G . AutG . AutG. . 1 is split. (ii) AutG.G/Aut G/.G,.G G . Sp(2n, Z) . (GL(m, Z) . (.)m). (iii) Aut G/.G,.G G/InnG . (. k )2n . (. k )2nm.