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The swap conjecture of Tennant and Turner. (English. Russian original) Zbl 0852.20025

Algebra Logic 34, No. 4, 249-257 (1995); translation from Algebra Logika 34, No. 4, 448-463 (1995).
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. R. F. Tennant and 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 \(n\geq 2\).

MSC:

20F05 Generators, relations, and presentations of groups
20E05 Free nonabelian groups
20F38 Other groups related to topology or analysis
20E36 Automorphisms of infinite groups
20F16 Solvable groups, supersolvable groups

Citations:

Zbl 0790.20053
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References:

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