Title : ( An upper bound for the index of the center in capable groups with finite cyclic derived subgroups )
Authors: Marzieh Chakaneh , Farangis Johari , Saeed Kayvanfar , Azam Kaheni ,
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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)`|^2 |ϕ(|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.
Keywords
, Schur\\\\\\\\\\\\\\\'s theorem, capable group, system normalizer
@article{paperid:1089925,
author = {Chakaneh, Marzieh and Johari, Farangis and Kayvanfar, Saeed and Azam Kaheni},
title = {An upper bound for the index of the center in capable groups with finite cyclic derived subgroups},
journal = {Quaestiones Mathematicae},
year = {2022},
volume = {45},
number = {6},
month = {June},
issn = {1607-3606},
pages = {875--888},
numpages = {13},
keywords = {Schur\\\\\\\\\\\\\\\'s theorem; capable group; system normalizer},
}
%0 Journal Article
%T An upper bound for the index of the center in capable groups with finite cyclic derived subgroups
%A Chakaneh, Marzieh
%A Johari, Farangis
%A Kayvanfar, Saeed
%A Azam Kaheni
%J Quaestiones Mathematicae
%@ 1607-3606
%D 2022
