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.

Mathematics Subject Classification (2020):

• Primary: 20E34
• Secondary: 20F16
• 20F18

Key words:

• Schur’s theorem
• capable group
• system normalizer