On exact products of two dihedral groups

TL;DR

This paper classifies all possible exact products of two dihedral groups of orders 2m and 2n, where m and n are odd integers greater than or equal to 3, expanding understanding of their subgroup structures.

Contribution

It provides a complete classification of exact products of two dihedral groups for all odd orders, a previously unexplored case in group theory.

Findings

Complete classification of exact products for dihedral groups of orders 2m and 2n

Identification of conditions for subgroup intersections and products

Extension of known results to all odd order cases

Abstract

An exact product of two finite groups $H$ and $K$ is a finite group $X$ which contains $H$ and $K$ as subgroups, satisfying $X=HK$ and $H\cap K=\{1_X\}$. In this paper, we provide a classification of the exact products of two dihedral groups of orders $2m$ and $2n$ for all odd numbers $m,n\geq 3$.