On the number of epi-, mono- and homomorphisms of groups
- Authors: Brusyanskaya E.K.1,2, Klyachko A.A.1,2
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Affiliations:
- Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
- Moscow Center for Fundamental and Applied Mathematics
- Issue: Vol 86, No 2 (2022)
- Pages: 25-33
- Section: Articles
- URL: https://journals.rcsi.science/1607-0046/article/view/142255
- DOI: https://doi.org/10.4213/im9139
- ID: 142255
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Abstract
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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About the authors
Elena Konstantinovna Brusyanskaya
Lomonosov Moscow State University, Faculty of Mechanics and Mathematics; Moscow Center for Fundamental and Applied Mathematicswithout scientific degree, no status
Anton Aleksandrovich 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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