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.

摘要

如果 G 组作为某个组H的中心因素组H/Z ( H )出现，则称该组G是有能力的。受 Isaacs [11] 的结果的启发，在Proc。阿米尔。数学。社会党。129 (10) (2001), pp. 2853-2859，我们证明如果G是具有奇数阶循环派生子群G ' 的有能力群，则 | G / Z ( G )| 划分|( G/L )′| ² φ (| L |)| L |，其中φ是欧拉函数，L 是G的下中心级数的最小项。此外，没有这样的有能力的非幂零群G持有 | G / Z ( G )| = | G ′| ² . 特别是，| G / Z ( G )| = | G ′| ²当且仅当G是幂零的。