On the solubilizer of an element in a finite group
Banafsheh Akbari, Costantino Delizia, Carmine Monetta
TL;DR
This work investigates the local structure of finite groups via the solubility graph $Γ_S(G)$ and its neighborhoods $Sol_G(x)$. It develops a framework based on long commutators $[u_1,\dots,u_k]$ and the lower central series $\gamma_k(G)$ to derive nilpotency criteria from the vanishing of commutators on $Sol_G(x)$, culminating in a complete characterization for $k=3$ that ties to $G$ being nilpotent of class at most $2$ when such a pinning element exists. For larger $k$, the authors describe stringent structural restrictions on possible insoluble counterexamples, including a unique minimal normal subgroup $K$ with factors isomorphic to $PSL(2,q)$ or $Alt(7)$. In parallel, they establish strong arithmetic constraints on the size of solubilizers, ruling out small prime-power cardinalities and deriving consequences for normalizers and specific group instances. Together, these results advance understanding of how local solubilizer data governs global group structure and nilpotency properties, with potential implications for algorithmic group analysis and graph-theoretic characterizations of finiteness.
Abstract
The solubility graph $Γ_S(G)$ associated with a finite group $G$ is a simple graph whose vertices are the elements of $G$, and there is an edge between two distinct vertices if and only if they generate a soluble subgroup. In this paper, we focus on the set of neighbors of a vertex $x$ which we call the solubilizer of $x$ in $G$, $\mathrm{Sol}_G(x)$, investigating both arithmetic and structural properties of this set.