Czechoslovak Mathematical Journal, online first, 11 pp.

# On decomposability of finite groups

## Ruifang Chen, Xianhe Zhao

#### Received April 21, 2016.   First published March 2, 2017.

Ruifang Chen (corresponding author), Xianhe Zhao, School of Mathematics and Information Science, Henan Normal University, No. 46, East of Construction Road, Xinxiang 453007, Henan, P. R. China, e-mail: fang119128@126.com, zhaoxianhe989@163.com

Abstract: Let \$G\$ be a finite group. A normal subgroup \$N\$ of \$G\$ is a union of several \$G\$-conjugacy classes, and it is called \$n\$-decomposable in \$G\$ if it is a union of \$n\$ distinct \$G\$-conjugacy classes. In this paper, we first classify finite non-perfect groups satisfying the condition that the numbers of conjugacy classes contained in its non-trivial normal subgroups are two consecutive positive integers, and we later prove that there is no non-perfect group such that the numbers of conjugacy classes contained in its non-trivial normal subgroups are 2, 3, 4 and 5.

Keywords: non-perfect group; \$G\$-conjugacy class; \$n\$-decomposable group

Classification (MSC 2010): 20E45, 20D10

DOI: 10.21136/CMJ.2017.0197-16

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