The swap conjecture of Tennant and Turner
If \(\gamma=(g_1,\dots,g_m)\) denotes a finite generating set of the group \(G\), then by an elementary swap one means a transition to another generating set \(\gamma'\) of \(m\) elements differing from \(\gamma\) by exactly one element. Two \(m\)-generating sets \(\gamma\), \(\gamma'\) of \(G\) are called swap equivalent if there is a sequence of elementary swaps leading from \(\gamma\) to \(\gamma'\). This is a wider notion than that of the Nielsen equivalence of two \(m\)-generating sets. \textit{R. F. Tennant} and \textit{E. C. Turner} in 1992 [Rocky Mt. J. Math. 22, No. 3, 1083-1095 (1992; Zbl 0790.20053)] conjectured that any finite generating sets \(\gamma\), \(\gamma'\) of a group \(G\) of the same cardinality, are swap equivalent. Groups for which this conjecture holds are called swap groups. In the same paper Tennant and Turner noted that finitely generated Abelian groups and Fuchsian groups are swap groups. Also swap groups are all the groups whose automorphisms are induced by automorphisms of the free group and finitely generated free nilpotent groups. The author proves that the free metabelian group of rank 3 is not a swap group. The author also examines the same problem for finitely generated modules over the ring \(\Lambda_n\) of Laurent polynomials in \(n\) commuting variables. In this case all the bases of the rank \(m\geq 3\) are swap equivalent, though for \(m=2\) there are infinitely many swap equivalence classes of bases over \(\Lambda_n\) for any \(n\geq 2\).
finitely generated free nilpotent groups
scientific article; zbMATH DE number 907281
scientific article