Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2016; 53(2): 479-486

Printed March 31, 2016

https://doi.org/10.4134/BKMS.2016.53.2.479

Copyright © The Korean Mathematical Society.

Finite groups with a cyclic norm quotient

Junxin Wang

Shanxi University of Finance and Economics

Abstract

The norm $N(G)$ of a group $G$ is the intersection of the normalizers of all the subgroups of $G$. In this paper, the structure of finite groups with a cyclic norm quotient is determined. As an application of the result, an interesting characteristic of cyclic groups is given, which asserts that a finite group $G$ is cyclic if and only if ${\rm Aut}(G)/P(G)$ is cyclic, where $P(G)$ is the power automorphism group of $G$.

Keywords: norm, cyclic group, power automorphism

MSC numbers: Primary 20D10, 20D20