arXiv subjects

Elena K. Brusyanskaya

Publications and source records attributed to Elena K. Brusyanskaya.

3 recordsLinked to original sources

arXiv2024

On the number of tuples of group elements satisfying a first-order formula

Our result contains as special cases the Frobenius theorem (1895) on the~number of solutions to the equation $x^n=1$ in a finite group and the Solomon theorem (1969) on the number of solutions in a group to systems of equations with fewer equations than unknowns. Instead of systems of equations, we consider arbitrary first-order formulae in the group language with constants. Our result substantially generalizes the Klyachko--Mkrtchyan theorem (2014) on this topic.

math.GR↗

arXiv2021

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

It is 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 investigate the number of homomorphisms satisfying some natural conditions such as injectivity or surjectivity. The simplest nontrivial corollary of our results is the following fact: {\it in any finite group, the number of generating pairs $(x,y)$ such that $x^3=1=y^5$, is a multiple of the greatest common divisor of 15 and the order of the group $[G,G]\cdot\{g^{15}\;|\;g\in G\}$.

math.GR↗

arXiv2019

What do Frobenius's, Solomon's, and Iwasaki's theorems on divisibility in groups have in common?

Our result contains as special cases the Frobenius theorem (1895) on the number of solutions to the equation $x^n=1$ in a group, the Solomon theorem (1969) on the number of solutions in a group to a system of equations having fewer equations than unknowns, and the Iwasaki theorem (1985) on roots of subgroups. There are other curious corollaries on groups and rings.

math.GR↗