Covering monolithic groups with proper subgroups

Given a finite non-cyclic group $G$, call $\sigma(G)$ the smallest number of proper subgroups of $G$ needed to cover $G$. Lucchini and Detomi conjectured that if a nonabelian group $G$ is such that $\sigma(G) < \sigma(G/N)$ for every non-trivial normal subgroup $N$ of $G$ then $G$ is \textit{monolithic}, meaning that it admits a unique minimal normal subgroup. In this paper we show how this conjecture can be attacked by the direct study of monolithic groups.Comment: I wrote this paper for the Proceedings of the conference "Ischia Group Theory 2012" (March, 26th - 29th 2012

Covering certain monolithic groups with proper subgroups

Given a finite non-cyclic group $G$, call $\sigma(G)$ the least number of proper subgroups of $G$ needed to cover $G$. In this paper we give lower and upper bounds for $\sigma(G)$ for $G$ a group with a unique minimal normal subgroup $N$ isomorphic to $A_n^m$ where $n \geq 5$ and $G/N$ is cyclic. We also show that $\sigma(A_5 \wr C_2)=57$.Comment: Communications in Algebra (2012

Covering certain Wreath Products with Proper Subgroups

For a non-cyclic finite group $X$ let $\sigma(X)$ be the least number of proper subgroups of $X$ whose union is $X$. Precise formulas or estimates are given for $\sigma(S \wr C_{m})$ for certain nonabelian finite simple groups $S$ where $C_m$ is a cyclic group of order $m$

Maximal irredundant families of minimal size in the alternating group

Let $G$ be a finite group. A family $\mathcal{M}$ of maximal subgroups of $G$ is called `irredundant' if its intersection is not equal to the intersection of any proper subfamily. $\mathcal{M}$ is called `maximal irredundant' if $\mathcal{M}$ is irredundant and it is not properly contained in any other irredundant family. We denote by \mbox{Mindim}(G) the minimal size of a maximal irredundant family of $G$. In this paper we compute \mbox{Mindim}(G) when $G$ is the alternating group on $n$ letters

On the number of conjugacy classes of a permutation group

We prove that any permutation group of degree $n \geq 4$ has at most $5^{(n-1)/3}$ conjugacy classes.Comment: 9 page

Factorizing a Finite Group into Conjugates of a Subgroup

For every non-nilpotent finite group $G$, there exists at least one proper subgroup $M$ such that $G$ is the setwise product of a finite number of conjugates of $M$. We define $\gamma_{\text{cp}}\left( G\right)$ to be the smallest number $k$ such that $G$ is a product, in some order, of $k$ pairwise conjugated proper subgroups of $G$. We prove that if $G$ is non-solvable then $\gamma_{\text{cp}}\left( G\right) \leq36$ while if $G$ is solvable then $\gamma_{\text{cp}}\left( G\right)$ can attain any integer value bigger than $2$, while, on the other hand, $\gamma_{\text{cp}}\left( G\right) \leq4\log_{2}\left\vert G\right\vert$.Comment: 14 page

Covers and Normal Covers of Finite Groups

For a finite non cyclic group $G$, let $\gamma(G)$ be the smallest integer $k$ such that $G$ contains $k$ proper subgroups $H_1,\dots,H_k$ with the property that every element of $G$ is contained in $H_i^g$ for some $i \in \{1,\dots,k\}$ and $g \in G.$ We prove that if $G$ is a noncyclic permutation group of degree $n,$ then $\gamma(G)\leq (n+2)/2.$ We then investigate the structure of the groups $G$ with $\gamma(G)=\sigma(G)$ (where $\sigma(G)$ is the size of a minimal cover of $G$) and of those with $\gamma(G)=2.