Induction of Characters and Finite $p$-Groups

Abstract

Let $G$ be a finite $p$-group, where $p$ is an odd prime number, $H$ be a subgroup of $G$ and $\theta\in \Irr(H)$ be an irreducible character of $H$. Assume also that $|G:H|=p^2$. Then the character $\theta^G$ of $ G$ induced by $\theta$ is either a multiple of an irreducible character of $G$, or has at least $\frac{p+1}{2}$ distinct irreducible constituents.