Abstract
We introduce intermediate commutators and study their degrees. We define \((q, \{\})\)-capable groups and prove that a group G is \((q, \{\})\)-capable if and only if \(Z^{\wedge }_{(q, \{\})}(G)=1\).
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Brown, R., Johnson, D.L., Robertson, E.F.: Some computations of non-abelian tensor products of groups. J. Algebra 111(1), 177–202 (1987). https://doi.org/10.1016/0021-8693(87)90248-1
Brown, R., Loday, J.L.: Van Kampen theorems for diagrams of spaces. Topology 26(3), 311–335 (1987). https://doi.org/10.1016/0040-9383(87)90004-8
Conduché, D., Rodríguez-Fernández, C.: Non-abelian tensor and exterior products modulo \(q\) and universal \(q\)-central relative extensions. J. Pure Appl. Algebra 78, 139–160 (1992). https://doi.org/10.1016/0022-4049(92)90092-T
Ellis, G.: Tensor products and \(q\)-crossed modules. J. London Math. Soc. 51(2), 243–258 (1995). https://doi.org/10.1112/jlms/51.2.243
Ellis, G., Rodríguez-Fernández, C.: An exterior product for the homology of groups with integral coefficients modulo \(q\). Cahiers Topologie Geom. Differ. Categ. 30(4), 339–344 (1989)
Gustafson, W.H.: What is the Probability that Two Group Elements Commute? Am. Math. Month. 80(9), 1031–1034 (1973)
Niroomand, P., Rezaei, R.: On the exterior degree of finite groups. Comm. Algebra 39, 335–343 (2011). https://doi.org/10.1080/00927870903527568
Niroomand, P., Russo, F.G.: On the tensor degree of finite groups. Ars Combinatoria 131, 273–283 (2017)
Rusin, D.J.: What is the probability that two elements of a finite group commute? Pacific J. Math. 82(1), 237–247 (1979)
The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.10.1; (2019), (https://www.gap-system.org)
Acknowledgements
We would like to thank the anonymous referee of this paper for many helpful comments and useful suggestions. This work was partially supported by DPI-UnB and FAPDF-Brazil.
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Communicated by Maria Manuel Clementino.
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Bastos, R., de Oliveira, R., Donadze, G. et al. q-Tensor and Exterior Centers, Related Degrees and Capability. Appl Categor Struct 31, 2 (2023). https://doi.org/10.1007/s10485-022-09701-0
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DOI: https://doi.org/10.1007/s10485-022-09701-0