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On the Saxl graph of a permutation group. (English) Zbl 07167592
Summary: Let \(G\) be a permutation group on a set \(\Omega\). A subset of \(\Omega\) is a base for \(G\) if its pointwise stabiliser in \(G\) is trivial. In this paper we introduce and study an associated graph \(\Sigma(G)\), which we call the Saxl graph of \(G\). The vertices of \(\Sigma(G)\) are the points of \(\Omega\), and two vertices are adjacent if they form a base for \(G\). This graph encodes some interesting properties of the permutation group. We investigate the connectivity of \(\Sigma(G)\) for a finite transitive group \(G\), as well as its diameter, Hamiltonicity, clique and independence numbers, and we present several open problems. For instance, we conjecture that if \(G\) is a primitive group with a base of size 2, then the diameter of \(\Sigma(G)\) is at most 2. Using a probabilistic approach, we establish the conjecture for some families of almost simple groups. For example, the conjecture holds when \(G = S_n\) or \(A_n\) (with \(n > 12\)) and the point stabiliser of \(G\) is a primitive subgroup. In contrast, we can construct imprimitive groups whose Saxl graph is disconnected with arbitrarily many connected components, or connected with arbitrarily large diameter.

MSC:
20B35 Subgroups of symmetric groups
20B05 General theory for finite permutation groups
20B15 Primitive groups
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
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