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The soluble graph of a finite group

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GRA2 - Groups, representations and applications: new perspectives

Let G be a finite insoluble group with soluble radical R(G). The vertices of the soluble graph of G, denoted Gamma_S(G), are labelled by the elements in G\R(G), with x and y adjacent if they generate a soluble subgroup of G. This is a natural generalisation of the widely studied commuting graph of G.   In this talk I will report on joint work with Andrea Lucchini and Daniele Nemmi, which establishes several new results on Gamma_S(G). Our main theorem states that this graph is always connected, with diameter at most 5. We can construct examples with diameter 4, so our upper bound on the diameter is close to best possible. I will explain some of the main ideas and I will present a number of open problems.

This talk is part of the Isaac Newton Institute Seminar Series series.

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