The Number of Group Homomorphisms from $D_m$ into $D_n$

TL;DR

This paper provides a novel elementary method to count the number of group homomorphisms from one dihedral group to another, a problem not previously documented in standard literature.

Contribution

It offers the first explicit formula for counting homomorphisms between dihedral groups, expanding the understanding of their algebraic structure.

Findings

Derived a formula for homomorphisms from D_m to D_n

Accessible to undergraduate students with basic group theory

Fills a gap in the literature on dihedral group homomorphisms

Abstract

Counting homomorphisms between cyclic groups is a common exercise in a first course in abstract algebra. A similar problem, accessible at the same level, is to count the number of group homomorphisms from a dihedral group of order $2m$ into a dihedral group of order $2n$. While the solution requires only elementary group theory, the result does not appear in the literature or in the usual texts. As the solution may be of interest, particularly to those teaching undergraduate abstract algebra, it is provided in this note.