arXiv subjects

Mikhail Kabenyuk

Publications and source records attributed to Mikhail Kabenyuk.

10 recordsLinked to original sources

arXiv2026

Dense Exact Factorizations of Countable Topological Groups

We prove that every countably infinite group endowed with a topology of countable weight in which every nonempty open set is infinite admits an exact factorization into two dense infinite subsets. More generally, every finite partial exact factorization with nonempty factors extends to an exact factorization of the whole group in which both factors meet every member of a prescribed at most countable family of infinite subsets in infinitely many points. This settles the countable case of a question of Protasov and Slobodianiuk. The proof uses the finite graph Ramsey theorem, invariant finitely additive probability measures on amenable groups, and a structural lemma on quadratic coverings based on Neumann's coset covering lemma.

math.GR↗

arXiv2026

Subindices of two-element subsets of finite groups

Let $A$ be a nonempty subset of a group $G$. A right subfactor relative to $A$ is a subset $B$ maximal under inclusion among those for which every element of $AB$ has a unique representation as $ab$, with $a\in A$ and $b\in B$. Left subfactors are defined analogously using $BA$. The infima and suprema of the cardinalities of right and left subfactors define four \emph{subindices}. A subset is called \emph{index-stable} if its four subindices coincide. For any infinite group $G$, every nonempty finite subset of $G$ is index-stable. For finite groups, the situation is more complicated. We prove that for a two-element subset of a finite group, the left and right subindices coincide pairwise. Let $A=\{x,y\}$ with $x\ne y$, and let $X$ be the cyclic subgroup generated by $x^{-1}y$. Then the lower subindex of $A$ is $|G:X|\cdot\lceil |X|/3\rceil$, and the upper subindex is $|G:X|\cdot\lfloor |X|/2\rfloor$. These formulas imply that $A$ is index-stable if and only if $|X|$ is one of $2,3,4,5,7$. For any two-element subset of a finite group $G$, the lower subindex is at least $\lceil |G|/3\rceil$, and the upper subindex is at most $\lfloor |G|/2\rfloor$. We establish necessary and sufficient conditions for equality in each of these bounds. All two-element subsets of a finite group are index-stable if and only if every element of the group has order $1$, $2$, $3$, $4$, $5$, or $7$. We give a complete structural classification of such groups.

math.GR↗

arXiv2026

Factorizations of finite groups

A finite group $G$ is called $k$-factorizable if for every ordered factorization $|G|=a_1\cdots a_k$ into integers each greater than $1$ there exist subsets $A_1,\dots,A_k\subseteq G$ such that $|A_i|=a_i$ for each $i$ and $G=A_1\cdots A_k$. The main results are as follows. 1. (Theorem 1.1) For every integer $k\geq3$ there exists a finite group $G$ such that $G$ is not $k$-factorizable. 2. (Theorem 1.2) Let $G$ be a finite group of order $4m$. If a Sylow $2$-subgroup of $G$ is elementary abelian and all involutions of $G$ are conjugate, then $G$ has no factorization of the form $G=ABC$ with $|A|=|C|=2$ and $|B|=m$. 3. Only $8$ groups of order at most $100$ fail to be $k$-factorizable for some $k$ (Theorem 1.3).

math.GR↗

arXiv2026

Factors in infinite groups

Let $G$ be a group and $A\subseteq G$ a non-empty subset. A right $s$-factor associated with $A$ is a maximal subset $U\subseteq G$ such that the product $AU$ is direct. The lower and upper $s$-indices $|G:A|^-$ and $|G:A|^+$ are defined as the minimum and the supremum of the cardinalities of such maximal sets $U$. The subset $A$ is called stable if $|G:A|^- = |G:A|^+$, and $G$ is called stable if every subset of $G$ is stable. Using a graph-theoretic reformulation in terms of Cayley graphs, we prove that every infinite group is unstable. Equivalently, for every infinite group $G$ there exists a subset $A\subseteq G$ for which maximal subsets $U$ with direct product $AU$ do not all have the same cardinality. This gives a negative answer to Question 21.58 of the Kourovka Notebook.

math.GR↗

arXiv2026

Factors in finite groups and well-covered graphs

We study a combinatorial property of subsets in finite groups that is analogous to the notion of independence in graphs. Given a group $G$ and a non-empty subset $A\subset G$, we define a (right) $s$-factor as a subset $B\subset G$ satisfying the following conditions: (i) Every element of $AB$ can be written uniquely as $ab$ with $a\in A$ and $b\in B$. (ii) $B$ is maximal (with respect to inclusion) with this property. For a finite group $G$, the upper and lower indices of $A$ are the sizes of the largest and smallest $s$-factors associated with $A$. A subset is called stable if its upper and lower indices coincide. A group is called stable if all its subsets are stable. We then explore the connection between $s$-factors in groups and maximal independent sets in graphs. Specifically, we show that $s$-factors in $G$ associated with $A$ correspond to maximal independent sets in a Cayley graph Cay($G$, $S$), where $S=A^{-1}A\setminus\{e\}$. Consequently, the upper and lower indices of $A$ are equal to the independence number and the independent domination number of the associated Cayley graph. The concepts of $s$-factors, subset indices in groups, stable subsets, and stable groups (under different names) were introduced by Hooshmand in 2020. Later, Hooshmand and Yousefian-Arani classified stable groups using computer calculations. Using the connection with graphs, we compute the upper and lower indices for various groups and their subsets. Furthermore, we prove a classification theorem describing all stable groups without relying on computer calculations.

math.GR↗

arXiv2025

A Note on Lagrange Subsets of Finite Groups

In a finite group, a subset is called a Lagrange subset if its size divides the group order, and a factor if it admits a complementary subset. We provide a new and comparatively direct proof of the classification of groups in which every Lagrange subset is a factor. We show that any nontrivial such group must be a cyclic group of prime order, the cyclic group of order 4, or an elementary abelian group of order 4, 8, or 9.

math.GR↗

arXiv2024

Minimal pentagulations of $n$-gons

A planar graph $G$ is called a pentagulation of an $n$-gon ($n\geq$ is an integer) if all faces of $G$ are pentagons, except one, which is an $n$-gon. A $3$-connected pentagulation $G$ of an $n$-gon is called minimal if it has the smallest number of pentagons among all such $3$-connected pentagulations. It is known that minimal pentagulations of the $3$-gon and $4$-gon contain 15 and 14 pentagons, respectively. We determined all minimal pentagulations of $n$-gons for all $n$ such that $3\leq n\leq 12$ using computer calculations. The calculations employed the plantri package, which generates all planar triangulations for a given number of vertices. We also present several open questions on this topic.

math.CO↗

arXiv2024

Complete factorizations of finite groups

Let $G$ be a group. The subsets $A_1,\ldots,A_k$ of $G$ form a complete factorization of group $G$ if if they are pairwise disjoint and each element $g\in G$ is uniquely represented as $g=a_1\ldots a_k$, with $a_i\in A_i$. We prove the following theorem: Let $G$ be a finite nilpotent group. If $|G|=m_1\ldots m_k$ where $m_1,\ldots,m_k$ are integers greater $1$ and $k\geq3$, then there exist subsets $A_1,\ldots,A_k$ of $G$ which form a complete factorization of group $G$ and $|A_i|=m_i$ for all $i=1,2,\ldots,k$. In addition, we give several examples of building complete factorization for some groups and formulate one open question.

math.GR↗

arXiv2024

Factorizations of simple groups of order 168 and 360

A finite group $G$ is called $k$-factorizable if for any factorization $|G|=a_1\cdots a_k$ with $a_i>1$ there exist subsets $A_i$ of $G$ with $|A_i|=a_i$ such that $G=A_1\cdots A_k$. We say that $G$ is \textit{multifold-factorizable} if $G$ is $k$-factorizable for any possible integer $k\geq2$. We prove that simple groups of orders 168 and 360 are multifold-factorizable and formulate two conjectures that the symmetric group $S_n$ for any $n$ and the alternative group $A_n$ for $n\geq6$ are multifold-factorizable.

math.GR↗

arXiv2022

Factorizations of groups of small order

Let $G$ be a finite group and let $A_1,\ldots,A_k$ be a collection of subsets of $G$ such that $G=A_1\ldots A_k$ is the product of all the $A_i$'s with $|G|=|A_1|\ldots|A_k|$. We write $G=A_1\cdot\ldots\cdot A_k$ and call this a $k$-fold factorization of $G$ of the form $(|A_1|,\ldots,|A_k|)$ or more briefly an $(|A_1|,\ldots,|A_k|)$-factorization of $G$. Let $k\geq2$ be a fixed integer. If $G$ has an $(a_1,\ldots,a_k)$-factorization, whenever $|G|=a_1\ldots a_k$ with $a_i>1$, $i=1,\ldots,k$, we say that $G$ is $k$-factorizable. We say that $G$ is multifold-factorizable if $G$ is $k$-factorizable for any possible integer $k\geq2$. In this paper we prove that there are exactly $6$ non-multifold-factorizable groups among the groups of order at most $60$. Here is their complete list: $A_4$, $(C_2\times C_2)\rtimes C_9$, $A_4\times C_3$, $(C_2\times C_2\times C_2)\rtimes C_7$, $A_5$, $A_4\times C_5$. Some related open questions are presented.

math.GR↗