arXiv subjects
Publications and source records attributed to Saveliy V. Skresanov.
Let $G$ be a finite group and let $H$ be a proper subgroup of $G$ of minimal index. By applying an old result of Y. Berkovich, we provide a polynomial algorithm for computing $|G : H|$ for a permutation group $G$. Moreover, we find $H$ explicitly if $G$ is given by a Cayley table. As a corollary, we get an algorithm for testing whether a finite permutation group acts on a tree or not.
"Kourovka notebook" contains the question due to L.G. Kovàcs (Problem 8.23): If the dihedral group $D$ of order 18 is a section of a direct product $X\times Y$, must at least one of $X$ and $Y$ have a section isomorphic to $D$? The goal of our short paper is to give the positive answer to this question provided that $X$ and $Y$ are locally finite. In fact, we prove even more: If a non-trivial semidirect product $D=A\rtimes B$ of a cyclic $p$-group $A$ and a group $B$ of order $q$, where $p$ and $q$ are distinct primes, lies in a locally finite variety generated by a set $\mathfrak{X}$ of groups, then $D$ is a section of a group from $\mathfrak{X}$.