‎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 thi...

Let G be a finite permutation group on Ω. An ordered sequence of elements Ω, (ω1,…,ωt), is an irredundant base for if the pointwise stabilizer G(ω1,…,ωt) trivial and no point fixed by its predecessors. If all bases have same size we say that IBIS group. In this paper show primitive IBIS, then it must almost simple, affine-type, or diagonal type. Moreover prove diagonal-type groups only isomorph...

We give a complete characterization of countable primitive groups in several settings including linear groups, subgroups of mapping class groups, groups acting minimally on trees and convergence groups. The latter category includes as a special case Kleinian groups as well as subgroups of word hyperbolic groups. As an application we calculate the Frattini subgroup in many of these settings, oft...