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Garonzi and Lucchini~\cite{GL} explored finite groups $G$ possessing a normal $2$-covering, where no proper quotient of $G$ exhibits such a covering. Their investigation offered a comprehensive overview of these groups, delineating that such groups fall into distinct categories: almost simple, affine, product action, or diagonal. In this paper, we focus on the family falling under the diagonal type. Specifically, we present a thorough classification of finite diagonal groups possessing a normal $2$-covering, with the attribute that no proper quotient of $G$ has such a covering., Comment: 9 pages

Given a permutation group $G$, the derangement graph of $G$ is the Cayley graph with connection set the derangements of $G$. The group $G$ is said to be innately transitive if $G$ has a transitive minimal normal subgroup. Clearly, every primitive group is innately transitive. We show that, besides an infinite family of explicit exceptions, there exists a function $f:\mathbb{N}\to \mathbb{N}$ such that, if $G$ is innately transitive of degree $n$ and the derangement graph of $G$ has no clique of size $k$, then $n\le f(k)$. Motivation for this work arises from investigations on Erd\H{o}s-Ko-Rado type theorems for permutation groups.

We classify all uniserial modules of the solvable Lie algebra $\mathfrak{g}=\langle x\rangle \ltimes V$, where $V$ is an abelian Lie algebra over an algebraically closed field of characteristic 0 and $x$ is an arbitrary automorphism of $V$.

We show that the permutation module over $\mathbb{C}$ afforded by the action of $Sp_{2m}(2^f)$ on its natural module is isomorphic to the permutation module over $\mathbb{C}$ afforded by the action of $Sp_{2m}(2^f)$ on the union of the right cosets of $O_{2m}^+(2^f)$ and $O_{2m}^-(2^f)$.