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An upper bound for the index of the center in capable groups with finite cyclic derived subgroups
Quaestiones Mathematicae ( IF 0.6 ) Pub Date : 2021-07-27 , DOI: 10.2989/16073606.2021.1906349
Marzieh Chakaneh 1 , Farangis Johari 1 , Saeed Kayvanfar 1 , Azam Kaheni 2
Affiliation  

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.



中文翻译:

具有有限循环派生子群的能力群中中心索引的上限

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 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。

更新日期:2021-07-27
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