Normal extensions and induced characters of 2-groups Mn

Abstract

Let $D_{n}$, $Q_{n}$ and $SD_{n}$ be the dihedral group, the generalized quaternion group and the semidihedral group of order $2^{n+1}$, respectively. Let $C_{n}$ be the cyclic 2-group of order $2^{n}$. As is well-known these four kinds of 2-groups play an important role in character theory of 2-groups. Let $\phi$ be a faithful irreducible character of $H=D_{n}$, $Q_{n}$, $SD_{n}$ or $C_{n}$. In [3] we determined all the 2-groups $G$ such that $H$ is a normal subgroup of $G$ and the induced character $\phi^{G}$ is irreducible. There exist other nonabelian 2-groups $M_{n}$ with a cyclic subgroup of index 2. All the faithful irreducible characters of $M_{n}$ are algebraically conjugate to each other as in $H$. The purpose of the paper is to determine all the 2-groups $G$ with a no rmal subgroup isomorphic to $M_{n}$ such that $\phi^{G}$ is irreducible for a faithful irreducible characters $\phi$ of the no rmal subgroup.