A classification of finite primitive IBIS groups with alternating socle

TL;DR

This paper classifies finite primitive IBIS groups with an alternating socle and proposes a conjecture for classifying all almost simple primitive IBIS groups, advancing the understanding of their structure.

Contribution

It provides a complete classification of primitive IBIS groups with alternating socle and introduces a conjecture for all almost simple primitive IBIS groups.

Findings

Classified finite primitive IBIS groups with alternating socle.

Proposed a conjecture for classifying all almost simple primitive IBIS groups.

Abstract

Let $G$ be a finite permutation group on $\Omega$. An ordered sequence $(\omega_1,\ldots,\omega_\ell)$ of elements of $\Omega$ is an irredundant base for $G$ if the pointwise stabilizer is trivial and no point is fixed by the stabilizer of its predecessors. If all irredundant bases of $G$ have the same cardinality, $G$ is said to be an IBIS group. Lucchini, Morigi and Moscatiello have proved a theorem reducing the problem of classifying finite primitive IBIS groups $G$ to the case that the socle of $G$ is either abelian or non-abelian simple. In this paper, we classify the finite primitive IBIS groups having socle an alternating group. Moreover, we propose a conjecture aiming to give a classification of all almost simple primitive IBIS groups.