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 soluble group satisfying some extra conditions, for example if the derived subgroup of G has odd order or is nilpotent.
Finite soluble groups satisfying the swap conjecture
LUCCHINI, ANDREA
2015
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 soluble group satisfying some extra conditions, for example if the derived subgroup of G has odd order or is nilpotent.
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