Geodesic
Finite soluble groups satisfying the swap conjecture.
Journal of Algebraic Combinatorics, Tome 42 (2015) no. 4, pp. 907-915.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

For a $d$-generated finite group $G$, we consider the graph $\Delta_d(G)$ (swap graph) in which the vertices are the ordered generating $d$-tuples and in which two vertices $(x_1,... ,x_d)$ and $(y_1,... ,y_d)$ are adjacent if and only if they differ only by one entry. It was conjectured by Tennant and Turner that $\Delta_d(G)$ 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.
Classification : 20D10, 20F05, 05C25
Keywords: generating graphs, swap conjecture, finite soluble groups, numbers of generators, graphs of generating tuples, connected graphs 
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     author = {Lucchini, Andrea},
     title = {Finite soluble groups satisfying the swap conjecture.},
     journal = {Journal of Algebraic Combinatorics},
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     publisher = {mathdoc},
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     number = {4},
     year = {2015},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JAC_2015__42_4_a10/}
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Lucchini, Andrea. Finite soluble groups satisfying the swap conjecture.. Journal of Algebraic Combinatorics, Tome 42 (2015) no. 4, pp. 907-915. http://geodesic.mathdoc.fr/item/JAC_2015__42_4_a10/