相关论文: On subgroups of R.Thompson's group F and other dia…

Let $G$ be a group. We define the coprime graph of subgroups of $G$, denoted by $\mathcal P(G)$, is a graph whose vertex set is the set of all proper subgroups of $G$, and two distinct vertices are adjacent if and only if the order of the…

Let $G$ be a finite abelian group, written additively, and $H$ a subgroup of~$G$. The \emph{subgroup sum graph} $\Gamma_{G,H}$ is the graph with vertex set $G$, in which two distinct vertices $x$ and $y$ are joined if $x+y\in…

A combinatorial map is a connected topological graph cellularly embedded in a surface. This monograph concentrates on the automorphism group of a map, which is related to the automorphism group of a Klein surface and a Smarandache manifold,…

We give an explicit handy (and cocycle-free) description of the groupoid of weak maps between two crossed-modules in terms of certain digrams of groups which we we call a {\em butterflies}. We define composition of butterflies and this way…

Let $G$ be 2-generated group. The generating graph $\Gamma(G)$ of $G$ is the graph whose vertices are the elements of $G$ and where two vertices $g$ and $h$ are adjacent if $G = \langle g, h \rangle.$ This definition can be extended to a…

For a connected graph, a path containing all vertices is known as \emph{Hamiltonian path}. For general graphs, there is no known necessary and sufficient condition for the existence of Hamiltonian paths and the complexity of finding a…

The ellipticity graph of a free group $F$ was defined by I. Kapovich and M. Lustig in order to study the outer automorphism group of $F$, which acts on this graph. The graph was constructed to be analogous to the curve complex of a surface.…

An $r$-dual tower of groups is a nested sequence of finite groups, like the symmetric groups, whose Bratteli diagram forms an $r$-dual graded graph. Miller and Reiner introduced a special case of these towers in order to study the Smith…

In this paper, we study the graph classification problem from the graph homomorphism perspective. We consider the homomorphisms from $F$ to $G$, where $G$ is a graph of interest (e.g. molecules or social networks) and $F$ belongs to some…

Let $F$ and $G$ be simple finite undirected graphs. A graph $G$ is called $F$-irregular if any two of its distinct vertices belong to different numbers of copies of $F$ in $G$. According to the strong conjecture about $F$-irregular graphs…

Let $G$ be a finite, non-abelian group of the form $G = A N$, where $A \leq G$ is abelian, and $N \trianglelefteq G$ is cyclic. We prove that the commuting graph $\Gamma(G)$ of $G$ is either a connected graph of diameter at most four, or…

Any finite group can be encoded as the automorphism group of an unlabeled simple graph. Recently Hartke, Kolb, Nishikawa, and Stolee (2010) demonstrated a construction that allows any ordered pair of finite groups to be represented as the…

In 1987, the second author of this paper reported his conjecture, all finite simple groups $S$ can be characterized uniformly using the order of $S$ and the set of element orders in $S$, to Prof. J. G. Thompson. In their communications,…

We describe standard forms for elements of the higher-dimensional Thompson groups $nV$ arising from gridding subdivision processes. These processes lead to standard normal form descriptions for elements in these groups, and sizes of these…

The semidirect product of a finitely generated group dual with the symmetric group can be described through so-called group-theoretical categories of partitions (covers only a special case; due to Raum--Weber, 2015) and skew categories of…

There are a variety of ways to associate directed or undirected graphs to a group. It may be interesting to investigate the relations between the structure of these graphs and characterizing certain properties of the group in terms of some…

We produce two separate algebraic descriptions of the isomorphism classes of the solvable subgroups of the group PLo(I) of piecewise-linear orientation-preserving homeomorphisms of the unit interval under the operation of composition, and…

This paper investigates the relationship between strata of abelian differentials and various mapping class groups afforded by means of the topological monodromy representation. Building off of prior work of the authors, we show that the…

The prime graph $\Gamma(G)$ of a finite group $G$ (also known as the Gruenberg-Kegel graph) has as its vertices the prime divisors of $|G|$, and $p\text-q$ is an edge in $\Gamma(G)$ if and only if $G$ has an element of order $pq$. Since…

The {\em distinguishing number} of a group $G$ acting faithfully on a set $V$ is the least number of colors needed to color the elements of $V$ so that no non-identity element of the group preserves the coloring. The {\em distinguishing…