Geodesic
On the number of epi-, mono- and homomorphisms of groups
Izvestiya. Mathematics , Tome 86 (2022) no. 2, pp. 243-251

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It is well known that the number of homomorphisms from a group $F$ to a group $G$ is divisible by the greatest common divisor of the order of $G$ and the exponent of $F/[F,F]$. We study the question of what can be said about the number of homomorphisms satisfying certain natural conditions like injectivity or surjectivity. A simple non-trivial consequence of our results is the fact that in any finite group the number of generating pairs $(x,y)$ such that $x^3=1=y^5$ is divisible by the greatest common divisor of fifteen and the order of the group $[G,G]\cdot\{g^{15}\mid g\in G\}$.
Keywords: number of homomorphisms, equations in groups, Frobenius' theorem, Solomon's theorem. 
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     title = {On the number of epi-, mono- and homomorphisms of  groups},
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E. K. Brusyanskaya; A. A. Klyachko. On the number of epi-, mono- and homomorphisms of  groups. Izvestiya. Mathematics , Tome 86 (2022) no. 2, pp. 243-251. http://geodesic.mathdoc.fr/item/IM2_2022_86_2_a1/