Nonsolvable groups with five character codegrees

Abstract

Let G be a finite group and the codegree of irreducible character χ of group G is defined as $|G:\text{ker}(\chi )|/\chi (1).$ In this paper, we prove that if a nonsolvable group G has exactly five character codegrees, then G is ismorphic to one of the following groups: $G/M\cong L_2(2^f),$ where M is the unique nontrivial normal subgroup of G with order $2^{2f};$ $L_2(q)$ (q odd); ${\text{PGL}}_2(q)$ (q odd) or $M_{10}=(L_2(9):2).$

Keywords:

• Character codegree
• nonsolvable group

MSC2010:

• 20C15
• 20D25

Acknowledgment

The authors are grateful to the referee for his/her valuable suggestions.

Additional information

Funding

The project is supported by NSFC [Grant Nos. 11701421 and 11871011, to Liu] and a grant from the Simons Foundation [No. 499532, to YY].