An Upper Bound for the Index of the Center in Capable Groups with Finite Cyclic Derived Subgroups

Abstract

A group G is said to be capable if it occurs as the central factor group H/Z(H) for some group H. Motivated by the results of Isaacs [11], in Proc. Amer. Math. Soc. 129(10) (2001), pp. 2853–2859, we show that if G is a capable group with cyclic derived subgroup G ′ of odd order, then |G/Z(G)| divides |(G/L) ′ | 2ϕ(|L|)|L|, in which ϕ is Euler’s function and L is the smallest term of the lower central series of G. Moreover, there is no such capable nonnilpotent group G that holds |G/Z(G)| = |G ′ | 2 . In particular, |G/Z(G)| = |G ′ | 2 if and only if G is nilpotent.