Related papers: Minimal cover groups

Finite covers are a technique for building new structures from simpler ones. The original motivation to study finite covers is in the Ladder theorem of Zilber which describes how totally categorical structures are built from strictly…

For a finite connected graph $\mathcal{E}$ with set of edges $E$, a finite $E$-generated group $G$ is constructed such that the set of relations $p=1$ satisfied by $G$ (with $p$ a word over $E\cup E^{-1}$) is closed under deletion of…

Given a finite group $G$, we say that $G$ has weak normal covering number $\gamma_w(G)$ if $\gamma_w(G)$ is the smallest integer with $G$ admitting proper subgroups $H_1,\ldots,H_{\gamma_w(G)}$ such that each element of $G$ has a conjugate…

A finite non-abelian group $H$ is hamiltonian if all of its subgroups are normal. We compute the minimal orders of graphs having a hamiltonian group as their automorphism group. The fixing number of a graph $\Gamma$ is the minimum…

Let \pi(G) denote the set of prime divisors of the order of a finite group G. The prime graph of G is the graph with vertex set \pi(G) with edges {p,q} if and only if there exists an element of order pq in G. In this paper, we prove that a…

Recent results of Qu and Tuarnauceanu describe explicitly the finite p-groups which are not elementary abelian and have the property that the number of their subgroups is maximal among p-groups of a given order. We complement these results…

A subgroup $H$ of a group $G$ is said to be an $IC\Phi$-subgroup of $G$ if $H \cap [H,G] \le \Phi(H)$. We analyze the structure of a finite group $G$ under the assumption that some given subgroups of $G$ are $IC\Phi$-subgroups of $G$. A new…

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…

In 2007, Zhi-Wei Sun defined a \emph{covering number} to be a positive integer $L$ such that there exists a covering system of the integers where the moduli are distinct divisors of $L$ greater than 1. A covering number $L$ is called…

The topology of a separable metrizable space $M$ is \emph{generated} by a family $\mathcal{C}$ of its subsets provided that a set $A\subseteq M$ is closed in $M$ if and only if $A\cap C$ is closed in $C$ for each $C\in \mathcal{C}$. The…

By [6], a minimal group $G$ is called $z$-minimal if $G/Z(G)$ is minimal. In this paper, we present the $z$-Minimality Criterion for dense subgroups with some applications to topological matrix groups. For a locally compact group $G$, let…

A group G that is not finitely generated can be written as the union of a chain of proper subgroups. The cofinality spectrum of G, written CF(S), is the set of regular cardinals lambda such that G can be expressed as the union of a chain of…

Every finite group $G$ has a normal series each of whose factors is either a solvable group or a direct product of nonabelian simple groups. The minimum number of nonsolvable factors attained on all possible such series is called the…

A finite group $G$ is called an F-group if for every $x, y \in G \setminus Z(G)$, $C(x) \leq C(y)$ implies that $C(x) = C(y)$. On the otherhand, two elements of a group are said to be $z$-equivalent or in the same $z$-class if their…

In this paper we generalize minimal $p$-divisible groups defined by Oort to $F$-crystal over an algebraically closed field of positive characteristic. We prove a structural theorem and give an explicit formula of the Frobenius endomorphism…

Let $\lambda(G)$ be the maximum number of subgroups in an irredundant covering of a finite group $G$. We prove that the finite groups with $\lambda(G)=|G|-t$, where $t\leq 5$, are solvable, and classify such groups.

For a finite group $G$, let $\text{rdim}(G)$ denote the smallest dimension of a faithful, complex linear representation of $G$. It is clear that $\text{rdim}(H)\leq \text{rdim}(G)$ for any subgroup $H$ of $G$. We consider $G$ with the…

We determine the finite groups $G$ in which every subset $A \subseteq G$ of cardinality dividing the order of $G$ is a \emph{factor}, i.e. has a complement $B \subseteq G$ of cardinality $|G|/|A|$ such that $G = A \cdot B$ or $G = B \cdot…

Let $G$ be a finite group and $\alpha(G)=\frac{|C(G)|}{|G|}$\,, where $C(G)$ denotes the set of cyclic subgroups of $G$. In this short note, we prove that $\alpha(G)\leq\alpha(Z(G))$ and we describe the groups $G$ for which the equality…

The topological group version of the celebrated Banach-Mazur problem asks wether every infinite topological group has a non-trivial separable quotient group. It is known that compact groups have infinite separable metrizable quotient…