Related papers: Intersection of conjugate solvable subgroups in sy…

Let $G$ be a finite group and $N(G)$ be the set of its conjugacy class sizes. In the 1980's Thompson conjectured that the equality $N(G)=N(S)$, where $Z(G)=1$ and $S$ is simple, implies the isomorphism $G\simeq S$. In a series of papers of…

There exist combable groups in which the conjugacy problem is unsolvable. The isomorphism problem is unsolvable for certain recursive sequences of finite presentations of combable groups.

Let $G$ be a finite group, and let $\kappa(G)$ be the probability that elements $g$, $h\in G$ are conjugate, when $g$ and $h$ are chosen independently and uniformly at random. The paper classifies those groups $G$ such that $\kappa(G) \geq…

We introduce two refinements of the class of $5/2$-groups, inspired by the classes of automorphism groups of configurations and automorphism groups of unit circulant digraphs. We show that both of these classes have the property that any…

In this paper we show that a finite nonabelian characteristically simple group G satisfying n = |\pi(G)|+2 if and only if G is isomorphic to A5, where n is the number of isomorphism classes of derived subgroups of G and \pi(G) is the set of…

Let G be a group. The intersection graph G(G) of G is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper nontrivial subgroups of G; and there is an edge between two distinct…

Given a finite nonabelian semisimple group $G$, we describe those groups that have the same holomorph as $G$, that is, those regular subgroups $N\simeq G$ of $S(G)$, the group of permutations on the set $G$, such that…

A classification is given for factorizations of almost simple groups with at least one factor solvable, and it is then applied to characterize $s$-arc-transitive Cayley graphs of solvable groups, leading to a striking corollary: Except the…

Let $M$ be a maximal subgroup of a finite group $G$ and $K/L$ be a chief factor such that $L\leq M$ while $K\nsubseteq M$. We call the group $M\cap K/L$ a $c$\ns section of $M$. And we define $Sec(M)$ to be the abstract group that is…

The solubility graph $\Gamma_S(G)$ associated with a finite group $G$ is a simple graph whose vertices are the elements of $G$, and there is an edge between two distinct vertices if and only if they generate a soluble subgroup. In this…

In this paper, we study the structures of finite groups using some arithmetic conditions on the sizes of real conjugacy classes. We prove that a finite group is solvable if the prime graph on the real class sizes of the group is…

A group $G$ with conjugation operation is a rack. We call such racks \emph{group racks}. In this paper we study finite group racks via their subrack lattices. Heckenberger, Shareshian, and Welker proved that the isomorphism type of the…

Let $G$ be a finite group and $\sigma_1(G)=\frac{1}{|G|}\sum_{H\leq G}\,|H|$. Under some restrictions on the number of conjugacy classes of (non-normal) maximal subgroups of $G$, we prove that if $\sigma_1(G)<\frac{117}{20}\,$, then $G$ is…

We prove that if a finite group $G$ contains a conjugacy class $K$ whose square is of the form $1 \cup D$, where $D$ is a conjugacy class of $G$, then $\langle K \rangle$ is a solvable proper normal subgroup of $G$ and we completely…

We prove that the solvable radical of a finite group G coincides with the set of elements y having the following property: for any x in G the subgroup of G generated by x and y is solvable. We present analogues of this result for finite…

We determine all maximal subgroups of the almost simple groups with socle $T=\PSL(2,q)$, that is, of all groups $G$ such that $\PSL(2,q)\leqslant G\leqslant\PGammaL(2,q)$, with $q\geq 4$.

In this paper we describe all the finite almost simple groups whose Gruenberg--Kegel graphs coincide with Gruenberg--Kegel graphs of finite solvable groups.

Let $n>0$ be an integer and $\mathcal{X}$ be a class of groups. We say that a group $G$ satisfies the condition $(\mathcal{X},n)$ whenever in every subset with $n+1$ elements of $G$ there exist distinct elements $x,y$ such that $<x,y>$ is…

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 paper, we prove that if $G$ is a finite group in which elements of the same order outside the center are conjugate, then $G$ is abelian or is isomorphic to $S_3$, the symmetry group of degree 3.