相关论文: Non-Solvable Graph of a Finite Group and Solvabili…

Given a finite group $G$ with a normal subgroup $N$, the simple graph $\Gamma_\textit{G}( \textit{N} )$ is a graph whose vertices are of the form $|x^G|$, where $x\in{N\setminus{Z(G)}}$, and $x^G$ is the $G$-conjugacy class of $N$…

A graph $G$ is perfectly divisible if, for every induced subgraph $H$ of $G$, either $V(H)$ is a stable set or admits a partition into two sets $X_1$ and $X_2$ such that $\omega(H[X_1]) < \omega(H)$ and $H[X_2]$ is a perfect graph. In this…

A graph $\Gamma$ is said to be symmetric if its automorphism group $\rm Aut(\Gamma)$ acts transitively on the arc set of $\Gamma$. In this paper, we show that if $\Gamma$ is a finite connected heptavalent symmetric graph with solvable…

T.C. Burness and S.D. Scott \cite{3} classified finite groups $G$ such that the number of prime order subgroups of $G$ is greater than $|G|/2-1$. In this note, we study finite groups $G$ whose subgroup graph contains a vertex of degree…

Let $\mathcal{C}$ be a class of groups. A group $X$ is said to be residually a $\mathcal{C}$-group (conjugacy $\mathcal{C}$-separable) if, for any elements $x,y \in X$ that are not equal (not conjugate in $X$), there exists a homomorphism…

In 1968, John Thompson proved that a finite group $G$ is solvable if and only if every $2$-generator subgroup of $G$ is solvable. In this paper, we prove that solvability of a finite group $G$ is guaranteed by a seemingly weaker condition:…

Let $G$ be a finite group, and $S$ be a subset of $G\setminus\{1\}$ such that $S=S^{-1}$. Suppose that $Cay(G,S)$ is the Cayley graph on $G$ with respect to the set $S$ which is the graph whose vertex set is $G$ and two vertices $a,b\in G$…

A graph $G$ is $H$-induced-saturated if $G$ is $H$-free but deleting any edge or adding any edge creates an induced copy of $H$. There are non-trivial graphs $H$, such as $P_4$, for which no finite $H$-induced-saturated graph $G$ exists. We…

we obtain a necessary condition for the character degree graph with all of its vertices are odd degree of a finite solvable group G.

A group G is a vGBS group if it admits a decomposition as a finite graph of groups with all edge and vertex groups finitely generated and free abelian. We prove that the multiple conjugacy problem is solvable between two n-tuples A and B of…

In this communication, the co-maximal subgroup graph $\Gamma(G)$ of a finite group $G$ is examined when $G$ is a finite nilpotent group, finite abelian group, dihedral group $D_n$, dicyclic group $Q_{2^n}$, and $p$-group. We derive the…

Let $X$ be a non-empty set of positive integers and $X^*=X\setminus \{1\}$. The divisibility graph $D(X)$ has $X^*$ as the vertex set and there is an edge connecting $a$ and $b$ with $a, b\in X^*$ whenever $a$ divides $b$ or $b$ divides…

In this article we generalize the theory of subgroup graphs of subgroups of free groups to finite index subgroups $H$ of finitely generated groups $G$. We study and prove various properties of $H$ in relation to its subgroup graph…

Given a formation $\mathfrak F$, we consider the graph whose vertices are the elements of $G$ and where two vertices $g,h\in G$ are adjacent if and only if $\langle g,h \rangle \notin\mathfrak F$. We are interested in the two following…

The spectrum $\omega(G)$ of a finite group $G$ is the set of orders of its elements. The following sufficient criterion of nonsolvability is proved: if among the prime divisors of the order of a group $G$, there are four different primes…

Denote by $\omega(G)$ the number of orbits of the action of $Aut(G)$ on the finite group $G$. We prove that if $G$ is a finite nonsolvable group in which $\omega(G) \leqslant 5$, then $G$ is isomorphic to one of the groups…

The subgroup generated by all solvable normal subgroups in a pseudo-finite group with the descending chain condition on centralizers up to finite index is solvable. Additionally, there is no finitely generated pseudo-finite group whose…

A group $G$ is invariably generated (IG) if there is a subset $S \subseteq G$ such that for every subset $S' \subseteq G$, obtained from $S$ by replacing each element with a conjugate, $S'$ generates $G$. $G$ is finitely invariably…

The commuting graph ${\Gamma(G)}$ of a group $G$ is the simple undirected graph with group elements as a vertex set and two elements $x$ and $y$ are adjacent if and only if $xy=yx$ in $G$. By eliminating the identity element of $G$ and all…

For a finite group $G$, the vertices of the prime graph $\Gamma(G)$ are the primes that divide $|G|$, and two vertices $p$ and $q$ are connected by an edge if there is an element of order $pq$ in $G$. Prime graphs of solvable groups have…