### Graphs encoding the generating properties of a finite group

Algebra and Geometry Seminar

6th June 2018, 2:30 pm – 3:30 pm

Howard House, 4th Floor Seminar Room

The generating graph Γ(G) of a finite group G is the graph defined on the elements of G in such a way that two distinct vertices are connected by an edge if and only if they generate G. It was defined by Liebeck and Shalev and has been further investigated by many authors. Many deep structural results about finite groups can be expressed in terms of the generating graph. In the first part of the talk we will survey some of these results.

Of course Γ(G) encodes significant information only when G is a 2-generator group. In the second part of the talk we will introduce and investigate a wider family of graphs which encode the generating property of G when G is an arbitrary finite group. More precisely, for every a, b ∈ N, we define a graph Γ_{a,b}(G) whose vertices correspond to the elements of G^a ∪ G^b and in which two tuples (x_1, . . . , x_a) and (y_1, . . . , y_b) are adjacent if and only if 〈x_1, . . . , x_a, y_1, . . . , y_b〉 = G. We will study several properties of these graphs (isolated vertices, loops, connectivity, diameter of the connected components) and we will investigate the relations between their properties and the group structure, with the aim of understanding which information about G are encoded by these graphs.

In a joint paper with P. Cameron and C. Roney-Dougal an equivalence relation ≡_m has been introduced, where two elements are equivalent if each can be substituted for the other in any generating set for G. This relation can be refined to a new sequence ≡^{(r)}_m of equivalence relations by saying that x ≡^{(r)}_m y if each can be substituted for the other in any r-element generating set. The relations ≡^{(r)}_m become finer as r increases. It is interesting to investigate the value ψ(G) of r at which they stabilise to ≡_m. Results about ≡_m, ≡^{(r)}_m and ψ(G) can be reformulated and reinterpreted in terms of properties of the graphs Γ_{1,b}(G).

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