Related papers: Finite groups in which every cyclic subgroup is th…

Let $G$ be a finite group and assume $p$ is a prime dividing the order of $G$. Suppose for any such $p$, that every two abelian $p$-subgroups of $G$ of equal order are conjugate. The structure of such a group $G$ has been settled in this…

We expound a concise construction of finite groups and groupoids whose Cayley graphs satisfy graded acyclicity requirements. Our acyclicity criteria concern cyclic patterns formed by coset-like configurations w.r.t. subsets of the generator…

The intersection graph of a group $G$ is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper non-trivial subgroups of $G$, and there is an edge between two distinct vertices $H$…

For each finite classical group $G$, we classify the subgroups of $G$ which act transitively on a $G$-invariant set of subspaces of the natural module, where the subspaces are either totally isotropic or nondegenerate. Our proof uses the…

The intersection growth of a group $G$ is the asymptotic behavior of the index of the intersection of all subgroups of $G$ with index at most $n$, and measures the Hausdorff dimension of $G$ in profinite metrics. We study intersection…

In this paper we introduce and study the relative cyclic subgroup commutativity degrees of a finite group. We show that there is a finite group with $n$ such degrees for all $n \in \mathbb{N}^*\setminus \lbrace 2\rbrace$ and we indicate…

In this paper, we consider various graphs, namely: power graph, cyclic graph, enhanced power graph and commuting graph, on a finite semigroup $S$. For an arbitrary pair of these four graphs, we classify finite semigroups such that the…

In this paper we give an algorithm to determine all finite matrix groups over a number field. Our algorithm is based on the representation theory of finite groups.

We construct the first example of a finitely-presented, residually-finite group that contains an infinite sequence of non-isomorphic finitely-presented subgroups such that each of the inclusion maps induces an isomorphism of profinite…

In this paper, we determine the finite groups with a Sylow $r$-subgroup contained in a unique maximal subgroup. The proof involves a reduction to almost simple groups, and our main theorem extends earlier work of Aschbacher in the special…

Let $G$ be a finite group. We let $\f{m}(G)$ and $\sig(G)$ denote the number of maximal subgroups of $G$ and the least positive integer $n$ such that $G$ is written as the union of $n$ proper subgroups, respectively. In this paper we…

A subgroup of a finite group G is said to be second maximal if it is maximal in every maximal subgroup of G that contains it. A question which has received considerable attention asks: can every positive integer occur as the number of the…

For a cyclic group $a$, define the atom of $a$ as the set of all elements generating $a$. Given any two elements $a,b$ of a finite cyclic group $G$, we study the sumset of the atom of $a$ and the atom of $b$. It is known that such a sumset…

We establish conditions under which the fundamental group of a graph of finite $p$-groups is necessarily residually $p$-finite. The technique of proof is independent of previously established results of this type, and the result is also…

Two classic results, due to K. Doerk and P. Hall respectively, establish the solvability of those finite groups all of whose maximal subgroups are supersolvable, and the solvability of finite groups in which all maximal subgroups have prime…

In this paper we introduce and study the concept of cyclic subgroup commutativity degree of a finite group $G$. This quantity measures the probability of two random cyclic subgroups of $G$ commuting. Explicit formulas are obtained for some…

In this note we study a class of finite groups for which the orders of subgroups satisfy a certain inequality. In particular, characterizations of the well-known groups $\mathbb{Z}_2\times\mathbb{Z}_2$ and $S_3$ are obtained.

We study the collection of group structures that can be realized as a group of rational points on an elliptic curve over a finite field (such groups are well known to be of rank at most two). We also study various subsets of this collection…

Let A be a finite non-singleton set. For |A|=2 we show that the partial clone consisting of all selfdual monotone partial functions on A is not finitely generated, while it is the intersection of two finitely generated maximal partial…

A cover of a finite non-cyclic group $G$ is a family $\mathcal{H}$ of proper subgroups of $G$ whose union equals $G$. A cover of $G$ is called minimal if it has minimal size, and irredundant if it does not properly contain any other cover.…