On the multiplicities of the character codegrees

Abstract

Let G be a finite group and ? be an irreducible character of G, the number cod(?) = jG : Let $G$ be a finite group and $\chi$ be an irreducible character of $G$, the number $ \cod(χ) = |G: \kernel(χ)|/χ(1) $ is called the codegree of $\chi$. Also, $ \cod(G) = \{ \cod(χ) \ | \ χ\in \Irr(G) \} $. For $d\in\cod(G)$, the multiplicity of $d$ in $G$, denoted by $m_G^{\prime }(d)$, is the number of irreducible characters of $G$ having codegree $d$. A finite group $G$ is called a $T_k^{\prime }$-group for some integer $k\ge 1$, if there exists $d_0\in\cod(G)$ such that $m_G^{\prime }(d_0)=k$ and for every $d\in\cod(G)-\{d_0\}$, we have $m_G^{\prime }(d)=1$. In this note we characterize finite $T_k^{\prime }$-groups completely, where $k\ge 1$ is an integer.