Related papers: Finite groups with a certain number of cyclic subg…

A finite group is said to be $n$-cyclic if it contains $n$ cyclic subgroups. For a finite group $G$, the ratio of the number of cyclic subgroups to the number of subgroups is known as the cyclicity degree of the group $G$ and is denoted by…

A classification of finite groups in which every 3-maximal subgroup is K-U-subnormal is given.

All finite simple groups are determined with the property that every Galois orbit on conjugacy classes has size at most 4. From this we list all finite simple groups $G$ for which the normalized group of central units of the integral group…

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.

Given a discrete (resp. profinite) group $G$, we define $NCC(G)$ to be the smallest number of cyclic (resp. procyclic) subgroups of $G$ whose conjugates cover $G$. In this paper we determine all residually finite discrete groups with finite…

We introduce and study the concept of cyclicity degree of a finite group $G$. This quantity measures the probability of a random subgroup of $G$ to be cyclic. Explicit formulas are obtained for some particular classes of finite groups. An…

In this paper it is shown that every finite cyclic group satisfies the CI-property for the class of balanced configurations.

A group $G$ is said to have dense solitary subgroups if each non-empty open interval in its subgroup lattice $L(G)$ contains a solitary subgroup. In this short note, we find all finite groups satisfying this property.

In this paper, we classify the finite simple groups with an abelian Sylow subgroup.

Given a finite group $G$, we denote by $L(G)$ the subgroup lattice of $G$ and by ${\cal CD}(G)$ the Chermak-Delgado lattice of $G$. In this note, we determine the finite groups $G$ such that $|{\cal CD}(G)|=|L(G)|-k$, $k=1,2$.

Groups, in which every subgroup containing some fixed primary cyclic subgroup has a complement, are investigated.

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…

We provide explicit identity bases for finite cyclic semigroups.

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.…

For a finite noncyclic group $G$, let $\Cyc(G)$ be a set of elements $a$ of $G$ such that $\langle a,b\rangle$ is cyclic for each $b$ of $G$. The noncyclic graph of $G$ is a graph with the vertex set $G\setminus \Cyc(G)$, having an edge…

This paper deals with combinatorial aspects of finite covers of groups by cosets or subgroups. Let $a_1G_1,...,a_kG_k$ be left cosets in a group $G$ such that ${a_iG_i}_{i=1}^k$ covers each element of $G$ at least $m$ times but none of its…

For a finite group $G$ let $\sigma(G)$ (the "sum" of $G$) be the least number of proper subgroups of $G$ whose set-theoretical union is equal to $G$, and $\sigma(G)=\infty$ if $G$ is cyclic. We say that a group $G$ is $\sigma$-elementary if…

Let $G$ be a finite group. A proper subgroup $H$ of $G$ is said to be large if the order of $H$ satisfies the bound $|H|^3 \ge |G|$. In this note we determine all the large maximal subgroups of finite simple groups, and we establish an…

The number of subgroups and the number of cyclic subgroups are natural combinatorial invariants of a finite group. We investigate how restrictions on these quantities, together with the number of distinct prime divisors of $|G|$, enforce…

We characterize finite $p$-groups $G$ of order up to $p^7$ for which the group of central automorphisms fixing the center element-wise is of minimum possibe order.