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Natalia V. Maslova

Publications and source records attributed to Natalia V. Maslova.

10 recordsLinked to original sources
arXiv2026

Generating the symmetric group by three prefix reversals

The cubic pancake graphs are Cayley graphs over the symmetric group $\mathrm{Sym}_n$ generated by three prefix reversals. There is the following open problem: characterize all the sets of three prefix reversals that generate $\mathrm{Sym}_n$. As the largest prefix reversal of length $n$ is always included in a triple, we give a complete solution of the problem when any of the two smallest or the two largest lengths but $n$ are included in a triple of prefix reversals. Moreover, some conditions implying a triple of prefix reversals does not generate $\mathrm{Sym}_n$ are considered. Computational results on the diameter and the girth of some cubic pancake graphs are presented, and conjectures for future research are formulated.

math.CO↗
arXiv2025

Aspects of the commuting graph

The commuting graph of a group $G$ is the graph whose vertices are the elements of $G$, two distinct vertices joined if they commute. Our purpose in this paper is twofold: we discuss the computational problem of deciding whether a given graph is the commuting graph of a finite group; we give a quasipolynomial algorithm, and a polynomial algorithm for the case when the group is an extra\-special p-group for p an odd prime; we give new results on the question of whether the commuting graph of a given group is a cograph or a chordal graph, two classes of graphs defined by forbidden subgraphs. The problems are not unrelated, since there are a number of cases where hard computational problems on graphs are easier when restricted to special classes of graphs; we conjecture that the recognition problem is polynomial for cographs and chordal graphs.

math.GR↗
arXiv2025

On arithmetical properties and arithmetical characterizations of finite groups

Arithmetical properties of a finite group are properties of the group which are defined by its arithmetical parameters such as the order of the group, the element orders and so on. In this paper, we discuss a number of results on arithmetical properties of finite groups and characterizations of finite groups by their arithmetical parameters obtained recently. This paper is based on the talk by the author given in frame of (WM)^2 - World Meeting for Women in Mathematics-2022.

math.GR↗
arXiv2025

On combinatorial properties of Gruenberg--Kegel graphs of finite groups

If $G$ is a finite group, then the spectrum $ω(G)$ is the set of all element orders of $G$. The prime spectrum $π(G)$ is the set of all primes belonging to $ω(G)$. A simple graph $Γ(G)$ whose vertex set is $π(G)$ and in which two distinct vertices $r$ and $s$ are adjacent if and only if $rs \in ω(G)$ is called the Gruenberg-Kegel graph or the prime graph of $G$. In this paper, we prove that if $G$ is a group of even order, then the set of vertices which are non-adjacent to $2$ in $Γ(G)$ form a union of cliques. Moreover, we decide when a strongly regular graph is isomorphic to the Gruenberg-Kegel graph of a finite group. Besides this, we prove that a complete bipartite graph with each part of size at least $3$ can not be isomorphic to the Gruenberg-Kegel graph of a finite group.

math.GR↗
arXiv2025

On characterization of groups by isomorphism type of Gruenberg-Kegel graph

The Gruenberg-Kegel graph (or the prime graph) $Γ(G)$ of a finite group $G$ is the graph whose vertex set is the set of prime divisors of $|G|$ and in which two distinct vertices $r$ and $s$ are adjacent if and only if there exists an element of order $rs$ in $G$. A group $G$ is recognizable by isomorphism type of Gruenberg--Kegel graph if for every group $H$ the isomorphism between $Γ(H)$ and $Γ(G)$ as abstract graphs (i.\,e. unlabeled graphs) implies that $G\cong H$. In this paper, we prove that finite simple exceptional groups of Lie type ${^2}E_6(2)$ and $E_8(q)$ for $q \in \{3, 4, 5, 7, 8, 9, 17\}$ are recognizable by isomorphism type of Gruenberg-Kegel graph.

math.GR↗
arXiv2023

On characterization by Gruenberg-Kegel graph of finite simple exceptional groups of Lie type

The Gruenberg-Kegel graph $Γ(G)$ of a finite group $G$ is the graph whose vertex set is the set of prime divisors of $|G|$ and in which two distinct vertices $r$ and $s$ are adjacent if and only if there exists an element of order $rs$ in $G$. A finite group $G$ is called almost recognizable (by Gruenberg-Kegel graph) if there is only finite number of pairwise non-isomorphic finite groups having Gruenberg-Kegel graph as $G$. If $G$ is not almost recognizable, then it is called unrecognizable (by Gruenberg--Kegel graph). Recently P.J. Cameron and the first author have proved that if a finite group is almost recognizable, then the group is almost simple. Thus, the question of which almost simple groups (in particular, finite simple groups) are almost recognizable is of prime interest. We prove that every finite simple exceptional group of Lie type, which is isomorphic to neither ${^2}B_2(2^{2n+1})$ with $n\geq1$ nor $G_2(3)$ and whose Gruenberg-Kegel graph has at least three connected components, is almost recognizable. Moreover, groups $ {^2}B_2(2^{2n+1})$, where $n\geq1$, and $G_2(3)$ are unrecognizable.

math.GR↗
arXiv2023

Criterion of unrecognizability of a finite group by its Gruenberg-Kegel graph

The Gruenberg-Kegel graph $Γ(G)$ associated with a finite group $G$ has as vertices the prime divisors of $|G|$, with an edge from $p$ to $q$ if and only if $G$ contains an element of order $pq$. This graph has been the subject of much recent interest; one of our goals here is to give a survey of some of this material, relating to groups with the same Gruenberg-Kegel graph. However, our main aim is to prove several new results. Among them are the following. - There are infinitely many finite groups with the same Gruenberg-Kegel graph as the Gruenberg-Kegel of a finite group $G$ if and only if there is a finite group $H$ with non-trivial solvable radical such that $Γ(G)=Γ(H)$. - There is a function $F$ on the natural numbers with the property that if a finite $n$-vertex graph whose vertices are labelled by pairwise distinct primes is the Gruenberg-Kegel graph of more than $F(n)$ finite groups, then it is the Gruenberg-Kegel graph of infinitely many finite groups. (The function we give satisfies $F(n)=O(n^7)$, but this is probably not best possible.) - If a finite graph $Γ$ whose vertices are labelled by pairwise distinct primes is the Gruenberg-Kegel graph of only finitely many finite groups, then all such groups are almost simple; moreover, $Γ$ has at least three pairwise non-adjacent vertices, and $2$ is non-adjacent to at least one odd vertex. - Groups whose power graphs, or commuting graphs, are isomorphic have the same Gruenberg-Kegel graph. - The groups ${^2}G_2(27)$ and $E_8(2)$ are uniquely determined by the isomorphism types of their Gruenberg-Kegel graphs. In addition, we consider groups whose Gruenberg-Kegel graph has no edges. These are the groups in which every element has prime power order, and have been studied under the name \emph{EPPO groups}; completing this line of research, we give a complete list of such groups.

math.GR↗
arXiv2018

On the Pronormality of Subgroups of Odd Index in Finite Simple Groups

A subgroup $H$ of a group $G$ is said to be {pronormal} in $G$ if $H$ and $H^g$ are conjugate in $\langle H, H^g \rangle$ for every $g \in G$. Some problems in finite group theory, combinatorics, and permutation group theory were solved in terms of pronormality. In 2012, E. Vdovin and the third author conjectured that the subgroups of odd index are pronormal in finite simple groups. In this paper we disprove their conjecture and discuss a recent progress in the classification of finite simple groups in which the subgroups of odd index are pronormal.

math.GR↗
arXiv2016

On the pronormality of subgroups of odd indices in finite simple symplectic groups

A subgroup $H$ of a group $G$ is said to be pronormal in $G$ if $H$ and $H^g$ are conjugate in $\langle H, H^g\rangle$ for every element $g \in G$. In [Sib. Math. J. 2015. Vol. 56, no. 6] we proved that subgroups of odd indeces are pronormal in many finite simple group. In [Proc. Steklov Inst. Math., to appear] we proved that a group $PSp_{6n}(q)$ with $q \equiv \pm 3 \pmod 8$ contains a nonpronormal subgroup of odd index. In this paper we prove that if $n \not \in \{2^m, 2^m(2^{2k}+1) \mid m, k \in \mathbb{N} \cup \{0\}\}$ then a group $PSp_{2n}(q)$ with $q \equiv \pm 3 \pmod 8$ contains a nonpronormal subgroup of odd index; if $n=2^m$ then any subgroup of odd index is pronormal in a group $PSp_{2n}(q)$, where $q \equiv \pm 3 \pmod 8$.

math.GR↗