Nonuniqueness of semidirect decompositions for semidirect products with directly decomposable factors and applications for dihedral groups

TL;DR

This paper investigates the nonuniqueness of semidirect decompositions in groups, especially for products with diagonal actions, and applies findings to dihedral groups to classify their decompositions.

Contribution

It provides new results on semidirect decompositions for products with diagonal actions and applies these to classify dihedral group decompositions.

Findings

Complete description of semidirect decompositions for dihedral groups

Identification of nonuniqueness in semidirect decompositions

Determination of minimal permutation degrees for dihedral groups

Abstract

Nonuniqueness of semidirect decompositions of groups is an insufficiently studied question in contrast to direct decompositions. We obtain some results about semidirect decompositions for semidirect products with factors which are nontrivial direct products. We deal with a special case of semidirect product when the twisting homomorphism acts diagonally on a direct product, as well as for the case when the extending group is a direct product. We give applications of these results in the case of generalized dihedral groups and classic dihedral groups $D_{2n}$. For $D_{2n}$ we give a complete description of semidirect decompositions and values of minimal permutation degrees.