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Blyth, Russell D.; Morse, Robert Fitzgerald

Computing the nonabelian tensor squares of polycyclic groups. (English) Zbl 1195.20035

J. Algebra 321, No. 8, 2139-2148 (2009).
Summary: We develop a theory for computing the nonabelian tensor square and related computations for finitely presented groups and specialize it to polycyclic groups. This theory provides a framework for making nonabelian tensor square computations for polycyclic groups and is the basis of an algorithm for computing the nonabelian tensor square for any polycyclic group.

Cited in 2 Reviews
Cited in 20 Documents

MSC:

20F16 Solvable groups, supersolvable groups
20F05 Generators, relations, and presentations of groups
20E22 Extensions, wreath products, and other compositions of groups
20J05 Homological methods in group theory
20-04 Software, source code, etc. for problems pertaining to group theory
68W30 Symbolic computation and algebraic computation

Keywords:

polycyclic groups; nilpotent groups; finitely presented groups; nonabelian tensor squares; algorithms

Software:

GAP; Magma; Polycyclic; NQ
Cite Review PDF
Full Text: DOI

References:

[1] Beuerle, James R.; Kappe, Luise-Charlotte, Infinite metacyclic groups and their non-abelian tensor squares, Proc. Edinb. Math. Soc. (2), 43, 3, 651-662 (2000) · Zbl 0985.20022
[2] Beyl, F. Rudolf; Felgner, Ulrich; Schmid, Peter, On groups occurring as center factor groups, J. Algebra, 61, 1, 161-177 (1979) · Zbl 0428.20028
[3] Blyth, Russell D.; Moravec, Primož; Morse, Robert Fitzgerald; Fine, Benjamin; Rosenberger, Gerhard; Spellman, Dennis, On the derived subgroup of the free nilpotent group and its applications, Aspects of Infinite Groups. Aspects of Infinite Groups, Algebra Discrete Math., 1 (2008)
[4] Blyth, Russell D.; Moravec, Primož; Morse, Robert Fitzgerald, On the nonabelian tensor squares of free nilpotent groups of finite rank, (Kappe, Luise-Charlotte; Magidin, Arturo; Morse, Robert Fitzgerald, Computational Group Theory and the Theory of Groups. Computational Group Theory and the Theory of Groups, Contemp. Math., vol. 470 (2008), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 27-44 · Zbl 1187.20037
[5] Blyth, Russell D.; Morse, Robert Fitzgerald; Redden, Joanne L., On computing the non-abelian tensor squares of the free 2-Engel groups, Proc. Edinb. Math. Soc. (2), 47, 2, 305-323 (2004) · Zbl 1074.20025
[6] Bosma, Wieb; Cannon, John; Playoust, Catherine, The Magma algebra system. I. The user language, Computational Algebra and Number Theory. Computational Algebra and Number Theory, London, 1993. Computational Algebra and Number Theory. Computational Algebra and Number Theory, London, 1993, J. Symbolic Comput., 24, 3-4, 235-265 (1997) · Zbl 0898.68039
[7] Brown, R.; Johnson, D. L.; Robertson, E. F., Some computations of nonabelian tensor products of groups, J. Algebra, 111, 1, 177-202 (1987) · Zbl 0626.20038
[8] Brown, Ronald; Loday, Jean-Louis, Van Kampen theorems for diagrams of spaces, Topology, 26, 3, 311-335 (1987), with an appendix by M. Zisman · Zbl 0622.55009
[9] R. Keith Dennis, In search of new “Homology” functors having a close relationship to \(K\)-theory, unpublished preprint; R. Keith Dennis, In search of new “Homology” functors having a close relationship to \(K\)-theory, unpublished preprint
[10] B. Eick, W. Nickel, Polycyclic - Computing with polycyclic groups, 2002. A GAP Package, see [16]; B. Eick, W. Nickel, Polycyclic - Computing with polycyclic groups, 2002. A GAP Package, see [16] · Zbl 1163.20022
[11] Eick, Bettina; Nickel, Werner, Computing the Schur multiplicator and the nonabelian tensor square of a polycyclic group, J. Algebra, 320, 2, 927-944 (2008) · Zbl 1163.20022
[12] Ellis, Graham, On the capability of groups, Proc. Edinb. Math. Soc. (2), 41, 3, 487-495 (1998) · Zbl 0999.20025
[13] Ellis, Graham, On the computation of certain homotopical-functors, LMS J. Comput. Math., 1, 25-41 (1998), (electronic) · Zbl 0914.18009
[14] Ellis, Graham; Leonard, Frank, Computing Schur multipliers and tensor products of finite groups, Proc. R. Ir. Acad. Sect. A, 95, 2, 137-147 (1995) · Zbl 0863.20010
[15] Ellis, Graham J., Tensor products and \(q\)-crossed modules, J. London Math. Soc. (2), 51, 2, 243-258 (1995) · Zbl 0829.20073
[16] The GAP Group, GAP - Groups, Algorithms, and Programming, Version 4.11 (2008)
[17] Holt, Derek F.; Eick, Bettina; O’Brien, Eamonn A., Handbook of Computational Group Theory, Discrete Math. Appl. (Boca Raton) (2005), Chapman & Hall/CRC: Chapman & Hall/CRC Boca Raton, FL · Zbl 1091.20001
[18] Kappe, Luise-Charlotte, Nonabelian tensor products of groups: The commutator connection, (Groups St. Andrews 1997 in Bath, II. Groups St. Andrews 1997 in Bath, II, London Math. Soc. Lecture Note Ser., vol. 261 (1999), Cambridge Univ. Press: Cambridge Univ. Press Cambridge), 447-454 · Zbl 1011.20035
[19] Lennox, John C.; Robinson, Derek J. S., The Theory of Infinite Soluble Groups, Oxford Math. Monogr. (2004), Clarendon Press/Oxford Univ. Press: Clarendon Press/Oxford Univ. Press Oxford · Zbl 1059.20001
[20] Aidan McDermott, The nonabelian tensor product of groups: Computations and structural results, PhD thesis, National Univ. of Ireland, Galway, February 1998; Aidan McDermott, The nonabelian tensor product of groups: Computations and structural results, PhD thesis, National Univ. of Ireland, Galway, February 1998 · Zbl 0959.20030
[21] Miller, Clair, The second homology group of a group; relations among commutators, Proc. Amer. Math. Soc., 3, 588-595 (1952) · Zbl 0047.25703
[22] W. Nickel, nq - Nilpotent Quotients of Finitely Presented Groups, 2003. A GAP Package, see [16]; W. Nickel, nq - Nilpotent Quotients of Finitely Presented Groups, 2003. A GAP Package, see [16]
[23] Rocco, N. R., On a construction related to the nonabelian tensor square of a group, Bol. Soc. Brasil. Mat. (N.S.), 22, 1, 63-79 (1991) · Zbl 0791.20020
[24] Rocco, N. R., A presentation for a crossed embedding of finite solvable groups, Comm. Algebra, 22, 6, 1975-1998 (1994) · Zbl 0819.20032
[25] Segal, Daniel, Polycyclic Groups, Cambridge Tracts in Math., vol. 82 (1983), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0516.20001
[26] Stammbach, Urs, Über die ganzzahlige Homologie von Gruppen, Expo. Math., 3, 4, 359-372 (1985) · Zbl 0593.20060
[27] Whitehead, J. H.C., A certain exact sequence, Ann. of Math. (2), 52, 51-110 (1950) · Zbl 0037.26101
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