1. Groups in which the co-degrees of the irreducible characters are distinct
- Author
-
Mahdi Ebrahimi
- Subjects
Finite group ,Algebra and Number Theory ,010102 general mathematics ,Group Theory (math.GR) ,010103 numerical & computational mathematics ,20C15, 05C25 ,01 natural sciences ,Combinatorics ,Set (abstract data type) ,Character (mathematics) ,FOS: Mathematics ,0101 mathematics ,Mathematics - Group Theory ,Mathematics - Abstract
Let $G$ be a finite group and let $\rm{Irr}(G)$ be the set of all irreducible complex characters of $G$. For a character $\chi \in \rm{Irr}(G)$, the number $\rm{cod}(\chi):=|G:\rm{ker}\chi|/\chi(1)$ is called the co-degree of $\chi$. The set of co-degrees of all irreducible characters of $G$ is denoted by $\rm{cod}(G)$. In this paper, we show that for a non-trivial finite group $G$, $|\rm{Irr}(G)|=|\rm{cod}(G)|$ if and only if $G$ is isomorphic to the cyclic group $\mathbb{Z}_2$ or the symmetric group $S_3$.
- Published
- 2021
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