The non-abelian tensor product of normal crossed submodules of groups

Document Type : Research Paper


Faculty of Mathematical Sciences, Shahid Beheshti University, Tehran 19839, Iran.



In this article, the notions of non-abelian tensor and exterior products of two normal crossed submodules of a given crossed module of groups are introduced and some of their basic properties are established. In particular, we investigate some common properties between normal crossed modules and their tensor products, and present some bounds on the nilpotency class and solvability length of the tensor product, provided such information is given at least on one of the normal crossed submodules.


[1] Arias, D. and Ladra, M., The precise center of a crossed module, J. Group Theory 12 (2009), 247-269.
[2] Bacon, M.R. and Kappe, L.C., The nonabelian tensor square of a 2-generator p-group of class 2, Arch. Math. (Basel) 61 (1993), 508-516.
[3] Bacon, M.R., Kappe, L.C., and Morse, R.F., On the nonabelian tensor square of a 2-Engel group, Arch. Math. (Basel) 69 (1997), 353-364.
[4] Brown, R., Johnson, D.L., and Robertson, E.F., Some computations of nonabelian tensor products of groups, J. Algebra 111 (1987), 177-202.
[5] Brown, R. and Loday, J.-L., Excision homotopique en basse dimension, C.R. Acad. Sci. Ser. I Math. Paris 298 (1984), 353-356.
[6] Brown, R. and Loday, J.-L., Van Kampen theorems for diagrams of spaces, Topology 26 (1987), 311-335.
[7] Carrasco, P., Cegarra, A.M., and Grandje'an, A.R., (Co)Homology of crossed modules, J. Pure Appl. Algebra 168 (2002), 147-176.
[8] Dennis, R.K., In search of new homology functors having a close relationship to K-theory, preprint, Cornell University, Ithaca, NY, 1976
[9] Donadze, G., Ladra, M., and Thomas, V.Z., On some closure properties of the non-abelian tensor product, J. Algebra 472 (2017), 399-413.
[10] Ellis, G., The non-abelian tensor product of finite groups is finite, J. Algebra 111 (1987), 203-205.
[11] Grandje'an, A.R., and Ladra, M., On totally free crossed modules, Glasgow Math. J. 40 (1998), 323-332.
[12] Grandje'an, A.R. and Ladra, M., H2(T; G; $sigma$) and central extensions for crossed modules, Proc. Edinburg Math. Soc. 42 (1999), 169-177.
[13] Ladra, M., and Grandje'an, A.R., Crossed modules and homology, J. Pure Appl. Algebra. 95 (1994), 41-55.
[14] Miller, C., The second homology of a group, Proc. Amer. Math. Sot. 3 (1952), 588-595.
[15] Mohammadzadeh, H., Shahrokhi, S., and Salemkar, A.R., Some results on stem covers of crossed modules, J. Pure Appl. Algebra 218 (2014), 1964-1972.
[16] Moravec, P., The non-abelian tensor product of polycyclic groups is polycyclic, J. Group Theory 10 (2007), 795-798.
[17] Nakaoka, I.N., Non-abelian tensor products of solvable groups, J. Group Theory 3 (2000), 157-167.
[18] Norrie, K.J., Crossed modules and analogues of group theorems, Thesis, King’s College, Univ. of London, London, 1987.
[19] Pirashvili, T., Ganea term for CCG-homology of crossed modules, Extracta Mathematicae 15 (2000), 231-235.
[20] Salemkar, A.R., Talebtash, S., and Riyahi, Z., The nilpotent multipliers of crossed modules, J. Pure Appl. Algebra 221 (2017), 2119-2131.
[21] Vieites, A.M. and Casas, J.M., Some results on central extensions of crossed modules, Homology, Homotopy Appl. 4 (2002), 29-42.
[22] Visscher, M.P., On the nilpotency class and solvability length of non-abelian tensor products of groups, Arch. Math. 73 (1999), 161-171.
[23] Whitehead, J.H.C., On adding relations to homotopy groups, Ann. of Math 42 (1942), 409-428.