On Finding the Number of Homomorphism From Q_8

This study investigates the number of homomorphisms from the quaternion group into various finite groups. Quaternion groups, denoted as Q8, possess unique algebraic properties that make them intriguing subjects for group theory inquiries. The research explores the enumeration of homomorphisms from Q8into specific finite groups, providing insights into the structural relationships between these groups. Here, we derive general formulae for counting the number of homomorphisms from quaternion group into each of quaternion group, dihedral group, quasi-dihedral group and modular group by using only elementary group theory