For a d-generated finite group G we consider the graph Δd(G) (swap graph) in which the vertices are the ordered generating d-tuples and in which two vertices (x1, ⋯ , xd) and (y1, ⋯ , yd) are adjacent if and only if they differ only by one entry. It was conjectured by Tennant and Turner that Δd(G) is a connected graph. We prove that this conjecture is true if G is a finite soluble group.
The swap graph of the finite soluble groups
DI SUMMA, MARCO;LUCCHINI, ANDREA
2016
Abstract
For a d-generated finite group G we consider the graph Δd(G) (swap graph) in which the vertices are the ordered generating d-tuples and in which two vertices (x1, ⋯ , xd) and (y1, ⋯ , yd) are adjacent if and only if they differ only by one entry. It was conjectured by Tennant and Turner that Δd(G) is a connected graph. We prove that this conjecture is true if G is a finite soluble group.
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