Luigi Iorio | Searcharxiv

arXiv2026

On the Hamiltonicity of generating graphs of almost simple groups

The generating graph $\Gamma(G)$ of a finite group $G$ has vertex set $G\setminus\{1\}$, and two distinct vertices are adjacent if and only if they generate $G$. Breuer, Guralnick, Lucchini, Maroti and Nagy [Bull. Lond. Math. Soc. 42 (2010), 621--633] conjectured that, for every finite group $G$ with at least four elements, $\Gamma(G)$ contains a Hamiltonian cycle if and only if every proper quotient of $G$ is cyclic. They proved their conjecture for sufficiently large almost simple groups with alternating socle and for all almost simple groups with sporadic socle. In this paper, we complete the asymptotic picture for almost simple groups by proving the conjecture for sufficiently large almost simple groups with socle of Lie type.

math.GR↗

arXiv2026

Finite groups with a large normalized sum of element orders

For a finite group $G$, let $ψ(G)$ be the sum of the orders of its elements, and define the corresponding normalized sum as $ψ'(G) := ψ(G)/ψ(\mathcal{C}_{|G|})$, where $\mathcal{C}_{|G|}$ is the cyclic group of the same order as $G$. Inspired by analogous criteria for the classes of soluble, supersoluble, and nilpotent groups, our main result establishes that if $ψ'(G)>ψ'(D_8) = \frac{19}{43}$, then $G$ belongs to the well-understood class of groups with a modular subgroup lattice, whose structure theory allows us to readily identify all groups satisfying this bound. Moreover, the equality case is fully settled. Finally, our arguments lead to a complete description of all groups satisfying $ψ'(G)> ψ'(A_4) = \frac{31}{77}$, thereby fully determining the groups covered by the supersolubility criterion of Baniasad Azad and Khosravi [Canad. Math. Bull. 65 (2022), 30--38], and thus providing a more complete answer to a corresponding conjecture of Tǎrnǎuceanu.

math.GR↗