Abstract
Let G be a group. We denote by \(\nu (G)\) a certain extension of the non-Abelian tensor square \([G,G^{\varphi }]\) by \(G \times G\). We prove that if G is a finite potent p-group, then \([G,G^{\varphi }]\) and the k-th term of the lower central series \(\gamma _k(\nu (G))\) are potently embedded in \(\nu (G)\) (Theorem A). Moreover, we show that if G is a potent p-group, then the exponent \(\exp (\nu (G))\) divides \(p \cdot \exp (G)\) (Theorem B). We also study the weak commutativity construction of powerful p-groups (Theorem C).
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References
Blyth, R.D., Fumagalli, F., Morigi, M.: Some structural results on the non-abelian tensor square of groups. J. Group Theory 13, 83–94 (2010)
Blyth, R.D., Morse, R.F.: Computing the nonabelian tensor square of polycyclic groups. J. Algebra 321, 2139–2148 (2009)
Brown, R., Johnson, D.L., Robertson, E.F.: Some computations of non-abelian tensor products of groups. J. Algebra 111, 177–202 (1987)
Brown, R., Loday, J.L.: Excision homotopique en basse dimension. C. R. Acad. Sci. Paris Sér. I(298), 353–356 (1984)
Brown, R., Loday, J.-L.: Van Kampen theorems for diagrams of spaces. Topology 26, 311–335 (1987)
Bueno, T.P., Rocco, N.R.: On the \(q\)-tensor square of a group. J. Group Theory 14, 785–805 (2011)
Dixon, J.D., du Sautoy, M.P.F., Mann, A., Segal, D.: Analytic pro-\(p\) Groups. Cambridge University Press, Cambridge (1991)
Eick, B., Nickel, W.: Computing the Schur multiplicator and the nonabelian tensor square of a polycyclic group. J. Algebra 320, 927–944 (2008)
Ellis, G., Leonard, F.: Computing Schur multipliers and tensor products of finite groups. Proc. R. Irish Acad. 95A, 137–147 (1995)
Fernández-Alcober, G.A., González-Sánches, J., Jaikin-Zapirain, A.: Omega subgroups of pro-\(p\) groups. Isr. J. Math. 166, 393–412 (2008)
The GAP Group: GAP—Groups, Algorithms, and Programming, Version 4.10.1. https://www.gap-system.org (2019)
González-Sánches, J., Jaikin-Zapirain, A.: On the structure of normal subgroups of potent \(p\)-groups. J. Algebra 276, 193–209 (2004)
Gupta, N., Rocco, N., Sidki, S.: Diagonal embeddings of nilpotent groups. Illinois J. Math. 30, 274–283 (1986)
Kappe, L.-C.: Nonabelian tensor products of groups: the commutator connection. In: Proc. Groups St. Andrews 1997 at Bath, London Math. Soc. Lecture Notes, vol. 261, pp. 447–454 (1997)
Lima, B.C.R., Oliveira, R.N.: Weak commutativity between two isomorphic polycyclic groups. J. Group Theory 19, 239–248 (2016)
Miller, C.: The second homology group of a group: relations among commutators. Proc. Am. Math. Soc. 3, 588–595 (1952)
Moravec, P.: Groups of prime power order and their nonabelian tensor squares. Isr. J. Math. 174, 19–28 (2009)
Nakaoka, I.N., Rocco, N.R.: A survey of non-abelian tensor products of groups and related constructions. Bol. Soc. Paran. Mat. 30, 77–89 (2012)
Rocco, N.R.: On weak commutativity between finite p-groups, p odd. J. Algebra 76, 471–488 (1982)
Rocco, N.R.: On a construction related to the non-abelian tensor square of a group. Bol. Soc. Brasil Mat. 22, 63–79 (1991)
Rocco, N.R.: A presentation for a crossed embedding of finite solvable groups. Comm. Algebra 22, 1975–1998 (1994)
Sidki, S.N.: On weak permutability between groups. J. Algebra 63, 186–225 (1980)
Acknowledgements
The authors wish to thank Professor Noraí Rocco for interesting discussions and the anonymous referee for his/her insightful comments. This work was partially supported by FAPDF-Brazil. Funding was provided by Fundação de Apoio à Pesquisa do Distrito Federal (BR) (Grant No. 0193.001344/2016) and Conselho Nacional de Desenvolvimento Científico e Tecnológico. The first and second authors were supported by DPI/UnB.
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Bastos, R., de Melo, E., Gonçalves, N. et al. Non-Abelian tensor square and related constructions of p-groups. Arch. Math. 114, 481–490 (2020). https://doi.org/10.1007/s00013-020-01449-0
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DOI: https://doi.org/10.1007/s00013-020-01449-0