Roman’kov, V. A. 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 \(n\geq 2\). Reviewer: S.Andreadakis (Athens) Cited in 5 Documents 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 Keywords:elementary swaps; Nielsen equivalence; finite generating sets; swap groups; automorphisms; finitely generated free nilpotent groups; free metabelian group of rank 3; finitely generated modules; Laurent polynomials Citations:Zbl 0790.20053 PDFBibTeX XMLCite \textit{V. A. Roman'kov}, Algebra Logic 34, No. 4, 249--257 (1995; Zbl 0852.20025); translation from Algebra Logika 34, No. 4, 448--463 (1995) Full Text: DOI EuDML References: [1] R. F. Tennant and E. C. Turner, ?The swap conjecture,?Rocky Mountain J. Math.,22, 1083-1095 (1992). · Zbl 0790.20053 · doi:10.1216/rmjm/1181072713 [2] R. C. Lyndon and P. Schupp,Combinatorial Group Theory, Springer-Verlag, New York (1977). · Zbl 0368.20023 [3] S. Andreadakis, ?On the automorphisms of free groups and free nilpotent groups,?Proc. London Math. Soc.,15, 239-268 (1965). · Zbl 0135.04502 · doi:10.1112/plms/s3-15.1.239 [4] S. Bachmuth, ?Induced automorphisms of free groups and free metabelian groups,?Trans. Am. Math. Soc.,122, 1-17 (1966). · Zbl 0133.28101 · doi:10.1090/S0002-9947-1966-0190212-1 [5] P. A. Linnell, ?Relation modules and augmentation ideals of finite groups,?J. Pure Appl. Alg.,22, 143-164 (1981). · Zbl 0476.20008 · doi:10.1016/0022-4049(81)90056-6 [6] J. S. Williams, ?Free presentations and relation modules of finite groups,?J. Pure Appl. Alg.,3, 203-217 (1973). · Zbl 0268.20025 · doi:10.1016/0022-4049(73)90010-8 [7] C. K. Gupta, N. D. Gupta, and V. A. Roman’kov, ?Primitivity in free groups and free metabelian groups,?Can. J. Math.,44, 516-523 (1992). · Zbl 0782.20031 · doi:10.4153/CJM-1992-033-0 [8] C. K. Gupta and N. D. Gupta, ?Lifting primitivity of free nilpotent groups,?Proc. Am. Math. Soc.,114, 617-621 (1992). · Zbl 0752.20012 · doi:10.1090/S0002-9939-1992-1088442-0 [9] S. Bachmuth, ?Automorphisms of free metabelian groups,?Trans. Am. Math. Soc.,118, 93-104 (1965). · Zbl 0131.02101 · doi:10.1090/S0002-9947-1965-0180597-3 [10] S. Bachmuth and H. Y. Mochizuki, ?Aut(F) ? Aut(F/F?) is surjective forF free of rank ? 4,?Trans. Am. Math. Soc.,292, 81-101 (1985). · Zbl 0575.20031 [11] V. A. Roman’kov, ?Automorphism groups of free metabelian groups,? inRelationship Problems in Abstract and Applied Algebras [in Russian], Computer Center SO AN SSSR, Novosibirsk (1985), pp. 53-80. [12] O. Chein, ?IA-automorphisms of free and free metabelian groups,?Comm. Pure Appl. Math.,21, 605-629 (1968). · Zbl 0157.05201 · doi:10.1002/cpa.3160210608 [13] S. Bachmuth and H. Y. Mochizuki, ?IA-automorphisms of the free metabelian group of rank 3,?J. Alg.,55, 106-115 (1978). · Zbl 0401.20033 · doi:10.1016/0021-8693(78)90194-1 [14] S. Bachmuth and H. Y. Mochizuki, ?The non-finite generation of Aut(G),G ? free metabelian of rank 3,?Trans. Am. Math. Soc.,270, 693-700 (1982). · Zbl 0482.20025 [15] V. A. Roman’kov, ?Primitive elements in free groups of rank 3,?Mat. Sb.,182, 1074-1085 (1991). · Zbl 0743.20035 [16] A. A. Suslin, ?AlgebraicK-theory and the norm-residue homomorphism,?Itogi Nauki Tekhniki,25, 115-208 (1984). [17] A. A. Suslin,?The structure of a special linear group over a polynomial ring,?Izv. Akad. Nauk SSSR,41, No. 2, 235-252 (1977). · Zbl 0354.13009 [18] S. Bachmuth and H. Y. Mochizuki, ?E 2 ?SL 2 for most Laurent polynomial rings,?Am. J. Math.,104, 1181-1189 (1982). · Zbl 0513.20038 · doi:10.2307/2374056 [19] V. A. Roman’kov, ?Residue matrix groups,? inRelationship Problems in Abstract and Applied Algebras [in Russian], Computer Center SO AN SSSR, Novosibirsk (1985), pp. 35-52. [20] N. D. Gupta,Free Group Rings, Cont. Math.,66, (1987). · Zbl 0641.20022 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.