Non abelian tensor products of groups and related constructions, and applications

Last updated, April 10, 2014

Some remarks on history

The earliest paper which uses a version of the nonabelian tensor square, and so a replacement of the commutatot map by a morphism, is surely [1] by Claire Miller. The next publication [2] defined a tensor product for a crossed module, and Abe Lue was annoyed he had not thought of the more symmetric definition given in [5-6].

It should be emphasised that a start for the work with Loday was a seminar I gave in Strasbourg in 1981 on the joint work with Philip Higgins; Jean-Louis immediately saw its relevance. Indeed, he had a conjecture which I saw as a triadic Hurewicz Theorem. But Higgins and I had deduced the Relative Hurewicz Theorem from a van Kampen type theorem. So we conjectured a van Kampen type theorem for his n-cat-groups. In 1982 it was realised that such a theorem would give rise to applications of a nonabelian tensor product, and this was part of papers [5-7]. This work was improved with the input of the preprints [3-4]. The problem of calculation arose out of writing the paper [6], once we had seen that the tensor square of finite groups was finite. Calculations for dihedral and quaternionic groups appeared in [6], and the general problem of calculation was put to David Johnson, leading to the publication [8].

A seminar I gave in Binghampton attracted the interest of L.-C. Kappe, and much work from her group.

Graham Ellis was a PhD student at Bangor, 1984-7, hence his interest in this area.


This version has been revised to be in chronological order, at least in terms of years, in order to show better the development of the area.

Among `related constructions' we include non abelian tensor products of other algebraic structures (see [17]), such as Lie algebras, since these were motivated by the construction for groups. Also included (see [21,74]) is the Peiffer product of groups which act on each other.

Suggestions for further entries or other comments are welcomed. See also the survey by L.-C. Kappe, listed as [68].

On my preprint page is a link to presentation on this topic for  a Colloquium in Goettingen, May 5, 2011.

This bibliography can be used with the brief Introduction (pdf file) or Introduction (html) to the history of this area, see also relation with the Blakers-Massey theorem (pdf file). See also presentations on preprint page

  1.  Miller, Clair, `The second homology of a group', Proc. American Math. Soc. 3 (1952) 588-595.
  2. A. S.-T. Lue, `The Ganea map for nilpotent groups', J. London Math. Soc. 14 (1976) 309-312.
  3. R.K. Dennis, `In search of new "Homology" functors having a close relationship to K-theory', preprint Cornell, 1976. (pdf file).
  4. R.K. Dennis, `Addendum to "In search of new "Homology" functors having a close relationship to K-theory"', handwritten preprint Cornell. (pdf file).
  5. R.Brown, J.-L.Loday, `Excision homotopique en basse dimension', C.R. Acad. Sci. Ser. I. Math. Paris, 298 (1984) 353-356.
  6. R.Brown, J.-L.Loday, `Van Kampen theorems for diagrams of spaces', Topology, 26, 311-335, 1987.
  7. R.Brown, J.-L.Loday, `Homotopical excision, and Hurewicz theorems, for $n$-cubes of spaces', Proc. London Math.Soc.(3) 54 (1987) 176-192.
  8. R.Brown, D.L.Johnson, E.F.Robertson, `Some computations of non-abelian tensor products of groups', J. Algebra, 111, 177-202, 1987.
  9. G.J. Ellis, `The non-abelian tensor product of finite groups is finite', J. Algebra, 111, 203-205, 1987.
  10. G.J. Ellis, `Non-Abelian exterior products of groups and exact sequences in the homology of groups', Glasgow Math. J., 29, 13-19, 1987.
  11. G.J. Ellis, `Non-Abelian exterior products of Lie algebras and an exact sequence in the homology of Lie algebras', J. Pure Appl. Algebra, 46, 111-115, 1987.
  12. R.Aboughazi, `Produit tensoriel du groupe d'Heisenberg', Bull. Soc. Math. France, 15, 95-106, 1987.
  13. N.D.Gilbert, `The non-Abelian tensor square of a free product of groups', Arch. Math., 48, 369-375, 1987.
  14. N. Rocco, `Non abelian tensor products under Engelian actions:an approach via a related construction', Preprint, University of Brasilia, 1987.
  15. D.L.Johnson, `The non-Abelian square of a finite split metacyclic group', Proc. Edinburgh Math. Soc., 30, 91-95, 1987.
  16. G.J. Ellis, `Multirelative algebraic $K$-theory: the group $K_{2}(\Lambda ,I_{1},\ldots ,I_{n})$ and related computations', J. Algebra, {\bf 112} (1988), 271-289. correction
  17. G.J. Ellis, `An eight term exact sequence in algebraic $K$-theory', {\it Bull. Lond. Math. Soc.} {\bf 20} (1988), 245-247.
  18. G.J. Ellis, `Higher dimensional crossed modules of algebras', {\it J. Pure Appl. Algebra} {\bf 52} (1988), 277-282.
  19. D. Guin, `Cohomologie et homologie non-abeliennes des groupes', J. Pure Applied Algebra, 50 (1988) 109-137.
  20. R.Brown, `Triadic Van Kampen theorems and Hurewicz theorems', Proc. Int. Conf. Algebraic Topology, Evanston, 1988 ed. M.Mahowald, Cont. Math. 96, 113-134, 1989. pdf
  21. G.J.Ellis and C. Rodriguez-Fernandez, `An exterior product for the homology of groups modulo $q$', Cah. Top. Géom. Diff. Cat. 30, 339-343, 1989.
  22. N.D.Gilbert, P.J.Higgins, `The non-Abelian tensor product of groups and related constructions', Glasgow Math. J., 31, 17-29, 1989.
  23. D.L.Johnson, `Noncancellation and nonabelian tensor squares', Group theory (Singapore, 1987), 405--408, de Gruyter, Berlin-New York, 1989.
  24. G.J. Ellis, `Relative derived functors and the homology of groups', { Cahiers Top. G\'eom. Diff. Cat\'eg.} {\bf 31} 2 (1990), 121-135.
  25. T. Hannebauer, `On non-abelian tensor squares of linear groups', Arch. Math. 55 (1990) 30-34.
  26. N. Rocco, `On a Construction Related to the nonabelian Tensor Square of a Group', Bol. Soc. Bras. Mat., 22 (1991), 63-79.
  27. R.Brown, `$q$-perfect groups and universal $q$-central extensions', Publicacions Mat. 34, 291-297, 1991.
  28. R.Brown, `Computing homotopy types using crossed $n$-cubes of groups', Proc. Adams Memorial Symposium on Algebraic Topology, Manchester 1990, Volume I, ed N.Ray and G. Walker, London Math. Soc. Lecture Note Series 175, Cambridge University Press, 187-210, 1991.
  29. G.J. Ellis, `A non-Abelian tensor product of Lie algebras', Glasgow Math. J., 33, 101-120, 1991.
  30. C.Rodr\'iguez-Fern\'andez and E.G.Rodeja-Fernandez, The exact sequence in the homology of groups with integral coefficients modulo $q$, associated to two normal subgroups, Proceedings I.C.T.M.'89, Cahiers Topologie Géom. Différentielle Catégoriques XXXII (1991), 113-129.
  31. Baues, Hans Joachim; Conduché, Daniel, `On the tensor algebra of a nonabelian group and applications', $K$-Theory 5 (1991/92), no. 6, 531--554.
  32. D. Conduché and C. Rodriguez-Fernandez, Non-abelian tensor and exterior products modulo $q$ and universal $q$-central relative extension, J. Pure Applied Algebra, {\bf 78} (1992), 139-160.
  33. G.J.Ellis, T.Hurley, F.Leonard, `Some computations of the non-abelian tensor product', 7pp, Galway Preliminary Report, 1992.
  34. M.R.Bacon, L.-C. Kappe, `The non-abelian square of a 2-generator $p$-group of class 2', Arch. Math., 61 (1993) 508-516.
  35. G.J. Ellis, `Crossed squares and combinatorial homotopy', Math. Z. 214 (1993) 93-110.
  36. N. Rocco, `A crossed embedding of groups and the computation of certain invariants of finite solvable groups', Matematica Contemporanea, vol.7, part II (1994) 19-24.
  37. M.R.Bacon, `On the non-abelian square of a nilpotent group of class 2', Glasgow Math. J. 36 (1994) 291-297.
  38. N. Rocco, `A presentation for a crossed embedding of finite solvable groups', Comm. in Alg., 22 (1994) 1975-1998.
  39. N.Inassaridze, `Non-abelian Homology of Groups', Bull. Georgian Acad. Sci., 150, No 1, 13-17, 1994.
  40. D. Guin, `Cohomologie des algebres de Lie croissees et K-theorie de Milnor additif', Ann. Inst. Fourier Grenoble,  45 (1995) 93-118.
  41. A.J.Duncan, G.J.Ellis, N.D.Gilbert, `A Mayer-Vietoris sequence in group homology and the decomposition of relation modules', Glasgow Math. J. 37  (1995) 159-171.
  42. G.J. Ellis, `Tensor products and q-crossed modules', J. London Math. Soc., (2) 51 (1995) 243-258.
  43. G.J.Ellis, F.Leonard, `Computing Schur multipliers and tensor products of finite groups', Proc. Royal Irish Acad., 95A (1995) 137-147.
  44. M. Hartl, `The non-abelian tensor square for groups of class 2', J. Algebra, 179 (1996) 416-440.
  45. N.Inassaridze, `Non-abelian Tensor Products and Non-abelian Homology of Groups', J. Pure Applied Algebra, 112, 191-205, 1996.
  46. N.Inassaridze, `Finiteness of Non-abelian Tensor Product of groups', Theory and Applications of categories, Vol. 2, No 5, 55-61, 1996.
  47. N.Inassaridze, `Non-abelian Tensor Products of Finite Groups with Non-compatible Actions', Bull. Georgian Acad. Sci.,154, No 1, 25-27, 1996.
  48. G.J. Ellis, `On the tensor square of a prime power group', {\it Arch. Math.} Vol. {\bf 66} (1996), 467-469.
  49. M.R. Bacon, L.-C. Kappe, R.F. Morse, `On the nonabelian tensor square of 2-Engel groups', Archiv der Mathematik 69 (1997) 353-364.
  50. N.Inassaridze, `Non-abelian Tensor Products of Precrossed Modules', Bull. Georgian Acad. Sci., 155, No 3, 1997.
  51. H.Inassaridze, `Non-abelian cohomology with coefficients in crossed bimodules', Georgian Mathematical Journal Vol.4, No.6, 509-922, 1997.
  52. H.Inassaridze, `Non-abelian cohomology of groups', Georgian Mathematical Journal Vol.4, No.4, 313-332, 1997.
  53. A. McDermott, The non abelian tensor product of groups: Computations and structural results, PhD Dissertation, National University of Ireland, Galway, 1998.
  54. G.J. Ellis, `The Schur multiplier of a pair of groups', {\it Appl. Categ. Structures}, 6 (1998), 355--371.
  55. G.J. Ellis, `A bound for the derived and Frattini subgroups of a prime-power group', {\it Proc. Amer. Math. Soc.}, 126 (1998), 2513-2523.
  56. N.Inassaridze, `Relationship of non-abelian tensor products and non-abelian homology of groups with Whitehead's gamma functor', Proceedings of A.Razmadze Mathematical Institute 117, 31-51, 1998.
  57. G.J. Ellis, `On the Schur multiplier of a quotient of a direct product of groups', {\it Bull. Australian Math. Soc.} 58 (1998), 495-499.
  58. H.Inassaridze and N.Inassaridze, `The Second and the Third Non-abelian Homology of Groups', Bull. Georgian Acad. Sci., 1998.
  59. G.J. Ellis, `On the computation of certain homotopical functors', {\it LMS Journal of Computation and Mathematics} 1 (1998) 25-41.
  60. H.Inassaridze and N.Inassaridze, `New descriptions of the non-abelian homology of groups', Bull. Georgian Acad. Sci., 157, No 2, 196-200, 1998.
  61. G.J.Ellis  and A.McDermott, `Tensor products of prime-power groups ', {\it J. Pure Appl. Algebra}, 132 (1998) 119-128.  
  62. E.Khmaladze, `Non-abelian tensor product of Lie algebras modulo q', Bull. Georgian Acad. Sci., 1998.
  63. Visscher, Matthew Peter; On the nonabelian tensor product of groups. Thesis (Ph.D.)–State University of New York at Binghamton. 1998. 90 pp.
  64. M.P Visscher, `On the nilpotency class and solvability length of non abelian tensor products of groups', Arch. Math. 73 (1999) 161-171.
  65. Gnedbaye, Allahtan V. A non-abelian tensor product of Leibniz algebras. Ann. Inst. Fourier (Grenoble) 49 (1999), no. 4, 1149--1177.
  66. Casas, J. M. `Universal central extension and the second invariant of homology of crossed modules in Lie algebras', Comm. Algebra 27 (1999), no. 8, 3811--3821.
  67. H.Inassaridze and N.Inassaridze, `Non-abelian homology of groups', K-Theory, 18 (1999) 1-17.
  68. E.Khmaladze, `Non-abelian tensor and exterior products of Lie algebras modulo q and related constructions', Bull. Georgian Acad. Sci., 1999.
  69. L.-C.Kappe, ` Non abelian tensor products of groups: the commutator connection' , Proceedings Groups St Andrews at Bath 1997, Lecture Notes LMS 261 (1999) 447-454.   pdf
  70. L.-C. Kappe, N. Sarmin and M.P Visscher, `Two generator two-groups of class two and their non abelian tensor squares', Glasgow Math. J.  41 (1999) 417-430.
  71. E.Khmaladze, `Non-abelian tensor and exterior products modulo q and universal q-central relative extensions of Lie algebras', Homology, Homotopy and Applications, 1 (1999) 187-204.
  72. J.R. Beuerle and L.-C. Kappe, `Infinite metacyclic groups and their non abelian tensor squares', Proc. Edinburgh Math Soc. 43 (2000) 651-662.
  73. A. Mutlu and T.Porter, `Freeness conditions for crossed squares and squared complexes', Special issues dedicated to Daniel Quillen on the occasion of his sixtieth birthday, Part IV. $K$-Theory 20 (2000), 345--368.
  74. N.D.Gilbert, `The low dimensional homology of crossed modules', Homology, Homotopy and Applications 2 (2000) 41-50.
  75. N.Inassaridze and E.Khmaladze, More about homological properties of precrossed modules, Homology, Homotopy and Applications 2 (2000) 105-114.
  76. I.N. Nakaoka, `Non abelian tensor products of solvable groups', J. Group Theory 3 (2000) 157--167.
  77. W.A. Bogley and N.D. Gilbert, `The homology of Peiffer productes of groups', New York J. Math. 6 (2000) 55-71.
  78. R. Kurdiani, "Nonabelian tensor product of restricted Lie algebras" Bull. Georgian Acad. Sci. 164 (2001), no. 1, 32-34.
  79. Inassaridze, Nick, On nonabelian tensor product modulo $q$ of groups. Comm. Algebra 29 (2001), no. 6, 2657-2687.
  80. Ellis, Graham, On the relation between upper central quotients and lower central series of a group. Trans. Amer. Math. Soc. 353 (2001), no. 10, 4219-4234.
  81. E.Khmaladze, `Homology of Lie algebras with $\Lambda/q\Lambda$ coefficients and exact sequences', Theory and Applications of Categories, 10 (2002) 113-126.
  82. R. Kurdiani and T. Pirashvili, `A Liebniz algebra structure on the second tensor power', J. Lie Theory, 12 (2002) 583-596.
  83. N.Inassaridze, E.Khmaladze and M.Ladra, Non-abelian tensor product of Lie algebras and its derived functors, Extracta Mathematicae 17 (2002)  281-288.
  84. N. H. Sarmin, `Infinite two-generator groups of class two and their non abelian tensor squares', International Journal of Mathematics and Mathematical Sciences, 32 (2002) 615-625.
  85. A. R. Grandjeán and M. P. López , `H2q(T,G,\partial) and q-perfect Crossed Modules', Applied Categorical Structures 11 (2) (2003) 171-184.
  86. Biddle, David P., and Kappe, Luise-Charlotte, On subgroups related to the tensor center. Glasg. Math. J. 45 (2003), no. 2, 323–332.
  87. E. Neher, `An introduction to universal central extensions of Lie superalgebras', in Groups, rings, Lie and Hopf algebras (St. John's, NF, 2001, 141-166, (St. John's, NF, 2001), Math. Appl., 555, Kluwer Acad, Publ, Dordrecht, 2003.
  88. D. Arias, M. Ladra, and A. R.-Grandjeán , `Universal central extensions of precrossed modules and Milnor's relative K2', Journal of Pure and Applied Algebra 184 (2003) 139-154.  
  89. Redden, Joanne Lynn; The nonabelian tensor square of the free 2-Engel group of rank n. Thesis (Ph.D.)–Saint Louis University. 2003. 163 pp.
  90. Bacon, Michael R.; Kappe, Luise-Charlotte, On capable p-groups of nilpotency class two. Special issue in honor of Reinhold Baer (1902–1979). Illinois J. Math. 47 (2003), no. 1-2, 49–62.
  91. D. Conduché, H. Inassaridze and N. Inassaridze, `Mod q cohomology and Tate-Vogel cohomology of groups', Journal of Pure and Applied Algebra 189 (2004) 61-87.
  92. N.Inassaridze, E.Khmaladze and M.Ladra, Non-abelian homology of Lie algebras, Glasgow Math. J. 46 (2004), 417-429.
  93. G.Donadze, N.Inassaridze, and T.Porter, n-Fold Cech derived functors and generalised Hopf type formulas, K-theory Preprint Archives: http://www.math.uiuc.edu/ K-theory/0624/
  94. Russell D. Blyth, Robert F. Morse, Joanne L. Redden, `On computing the non-abelian square of the free 2-Engel groups', Proceedings of the Edinburgh Mathematical Society, 47  (2004) 305-323.
  95. G. Guerard, `Produit tensoriel non abeliens, relations entre commutateurs et homologie des groupes', These doctorale, Universite de Rennes, May, 2005.  
  96. Morse, Robert Fitzgerald , `Advances in Computing the Nonabelian Tensor Square of Polycyclic Groups', Irish Math. Soc. Bulletin 56 (2005), 115-123.
  97. Muro, Fernando Suspensions of crossed and quadratic complexes, co-$H$-structures and applications. Trans. Amer. Math. Soc. 357 (2005), no. 9, 3623-3653
  98. Moravec, Primož On nonabelian tensor analogues of 2-Engel conditions. Glasg. Math. J. 47 (2005), no. 1, 77-86.
  99. Donadze, Guram; Inassaridze, Nick; Porter, Timothy $N$-fold Cech derived functors and generalised Hopf type formulas. $K$-Theory 35 (2005), no. 3-4, 341-373 (2006).
  100. Bak, A.; Donadze, G.; Inassaridze, N.; Ladra, M. Homology of multiplicative Lie rings. J. Pure Appl. Algebra 208 (2007), no. 2, 761-777.
  101. Moravec, Primož The non-abelian tensor product of polycyclic groups is polycyclic. J. Group Theory 10 (2007), no. 6, 795-798.
  102. Donadze, G.; Ladra, M. More on five commutator identities. arx:0703633.
  103. Baues, H.-J.; Jibladze, M.; Pirashvili, T. Quadratic algebra of square groups. Adv. Math. 217 (2008), no. 3, 1236-1300.
  104. Moravec, P., `The exponents of nonabelian tensor products of groups', J. Pure Appl. Algebra 212 (2008), no. 7, 1840-1848.
  105. Eick, Bettina; Nickel, Werner Computing the Schur multiplicator and the nonabelian tensor square of a polycyclic group. J. Algebra 320 (2008), no. 2, 927--944
  106. Moghaddam, Mohammad Reza R.; Salemkar, Ali Reza; Karimi, Taghi Some inequalities for the order of the Schur multiplier of a pair of groups. Comm. Algebra 36 (2008), no. 7, 2481-2486.
  107. Blyth, Russell D.; Moravec, Primo; Morse, Robert Fitzgerald On the nonabelian tensor squares of free nilpotent groups of finite rank. Computational group theory and the theory of groups, 27-43, Contemp. Math., 470, Amer. Math. Soc., Providence, RI, 2008.
  108. Moravec, P., The exponents of nonabelian tensor products of groups. J. Pure Appl. Algebra 212 (2008), no. 7, 1840-1848.
  109. R. Brown and R. Sivera, `Algebraic colimit calculations in homotopy theory using fibred and cofibred categories',  Theory and Applications of Categories, 22 (2009) 222-251.
  110. Blyth, Russell D.; Morse, Robert Fitzgerald, Computing the nonabelian tensor squares of polycyclic groups. J. Algebra 321 (2009), no. 8, 2139–2148.
  111. Moravec, Primo, Groups of prime power order and their nonabelian tensor squares. Israel J. Math. 174 (2009), 19-28.
  112. Nakaoka, Irene N.; Rocco, Nora; R. A note on semidirect products and nonabelian tensor products of groups. Algebra Discrete Math. 2009, no. 3, 77–84.
  113. S. H. Jafari, P. Niroomand, A. Erfanian, The Non-Abelian Tensor Square and Schur multiplier of Groups of Orders $p^2q$, $pq^2$ and $p^2qr, arXiv:0912.4000v1 [math.GR].
  114. Moravec, Primo, Powerful actions and non-abelian tensor products of powerful p-groups. J. Group Theory 13 (2010), no. 3, 417-427 .
  115. Russo, Francesco, Nonabelian tensor product of soluble minimax groups. Computational group theory and the theory of groups, II, 179-183, Contemp. Math., 511, Amer. Math. Soc., Providence, RI, 2010.
  116. Thomas, V.Z., `The nonabelian tensor product of finite groups is finite: a homology free proof', Glasgow J. Mathematics, 52 (2010), no. 3, 473-477.
  117. Ellis, G.J., About HAP: Third Homotopy Groups Of Suspensions Of Classifying Spaces: http://hamilton.nuigalway.ie/Hap/www/SideLinks/About/aboutTensorSquare.html
  118. Eick, Bettina, Nickel, Werner, `Computing the Schur multiplicator and the nonabelian tensor square of a polycyclic group', J. Algebra 320 (2008), no. 2, 927-944.
  119. Thomas, Viji Z.; The box-tensor product: A generalization of the nonabelian tensor product. Thesis (Ph.D.) State University of New York at Binghamton. 2010. 68 pp.
  120. Rashid, S.; Sarmin, N. H.; Erfanian, A.; Mohd Ali, N. M. On the nonabelian tensor square and capability of groups of order p2q. Arch. Math. (Basel) 97 (2011), no. 4, 299-06.
  121. Moravec, P., On the Schur multipliers of finite p-groups of given coclass. Israel J. Math. 185 (2011), 189-205.
  122. Ahmad Erfanian, Francesco G. Russo, Nor Haniza Sarmin, Some considerations on the nonabelian tensor square of crystallographic groups, Asian European J. Math. (2011), 271 - 282 . arXiv:0911.5604
  123. Bueno, Ticianne P.; Rocco, N. R. On the q-tensor square of a group. J. Group Theory 14 (2011), no. 5, 785-805.
  124. Dietrich, Heiko; Moravec, Primoz, On the autocommutator subgroup and absolute centre of a group. J. Algebra 341 (2011), 150-157.
  125. Francesco G. Russo A note on the invariance in the nonabelian tensor product arXiv:1007.1323 4p.
  126. Manuel Ladra, Viji Z. Thomas , Two generalizations of the nonabelian tensor product, J. Algebra 369 (2012), 96-113 . arXiv:1106.2136 19p.
  127. Parvizi, Mohsen; Niroomand, Peyman On the structure of groups whose exterior or tensor square is a p-group. J. Algebra 352 (2012), 347-353.
  128. Jafari, S. H. A bound on the order of non-abelian tensor square of a prime-power group. Comm. Algebra 40 (2012), no. 2, 528-530.
  129. Moghaddam, Mohammad Reza R.; Niroomand, Peyman. Some properties of certain subgroups of tensor squares of p-groups. Comm. Algebra 40 (2012), no. 3, 1188 - 1193.
  130. Otera, Daniele Ettore; Russo, Francesco G.; Tanasi, Corrado. Some algebraic and topological properties of the nonabelian tensor product. Bull. Korean Math. Soc. 50 (2013), no. 4, 10691077.
  131. Rashid, S.; Sarmin, N. H.; Erfanian, A.; Ali, N. M. Mohd; Zainal, R. On the nonabelian tensor square and capability of groups of order 8q. Indag. Math. (N.S.) 24 (2013), no. 3, 581588.

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