## Hokkaido Mathematical Journal

- Hokkaido Math. J.
- Volume 30, Number 1 (2001), 163-176.

### Normal extensions and induced characters of 2-groups M_{n}

#### 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.

#### Article information

**Source**

Hokkaido Math. J., Volume 30, Number 1 (2001), 163-176.

**Dates**

First available in Project Euclid: 22 October 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.hokmj/1350911929

**Digital Object Identifier**

doi:10.14492/hokmj/1350911929

**Mathematical Reviews number (MathSciNet)**

MR1815005

**Zentralblatt MATH identifier**

0991.20007

**Subjects**

Primary: 20C15: Ordinary representations and characters

Secondary: 20D15: Nilpotent groups, $p$-groups

**Keywords**

2-group induced character group extension

#### Citation

IIDA, Youichi. Normal extensions and induced characters of 2-groups M n. Hokkaido Math. J. 30 (2001), no. 1, 163--176. doi:10.14492/hokmj/1350911929. https://projecteuclid.org/euclid.hokmj/1350911929