On the number of epi-, mono- and homomorphisms of groups

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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 $[G,G]\cdot\{g^{15}\mid g\in G\}$.

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作者简介

Elena Brusyanskaya

Lomonosov Moscow State University, Faculty of Mechanics and Mathematics; Moscow Center for Fundamental and Applied Mathematics

without scientific degree, no status

Anton Klyachko

Lomonosov Moscow State University, Faculty of Mechanics and Mathematics; Moscow Center for Fundamental and Applied Mathematics

Email: anton.klyachko@gmail.com
Candidate of physico-mathematical sciences, no status

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