Hokkaido Mathematical Journal

Normal extensions and induced characters of 2-groups Mn

Youichi IIDA

Full-text: Open access

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


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