Groups with vanishing class size \(p\) (Q2285738)

By definition an element $g$ of a group $G$ is said to be vanishing if there is an irreducible character $X$ of $G$ such that $X(g)=0$, in this case the conjugacy class of $g$ is called a vanishing class.The main result of this paper states that if $G$ is a finite group and the vanishing class of $G$ all have size $p$, for some prime $p$, then (1) $G=PXH$ where $P$ is a $p$-group whose conjugacy classes have size 1 or $p$ and $H$ is a $p'$-group. (2) $G/Z(G)$ is a Frobenius group with Frobenius kernel of order $p$.