On some subgroups associated with the tensor square of a group

Document Type : Research Paper


1 Department of Maths,birjand university

2 Department of math, birjand university

3 Islamic Azad University, Neyshabur branch


‎In this paper we present some results about subgroup which is‎ ‎generalization of the subgroup $R_{2}^{\otimes}(G)=\{a\in‎ ‎G|[a,g]\otimes g=1_{\otimes},\forall g\in G\}$ of right‎ ‎$2_{\otimes}$-Engel elements of a given group $G$‎. ‎If $p$ is an‎ ‎odd prime‎, ‎then with the help of these results‎, ‎we obtain some‎ ‎results about tensor squares of p-groups satisfying the law‎ ‎$[x,g,y]\otimes g=1_{\otimes}$‎, ‎for all $x‎, ‎g‎, ‎y\in G$‎. ‎In‎ ‎particular p-groups satisfying the law $[x,g,y]\otimes‎ ‎g=1_{\otimes}$ have abelian tensor squares‎. ‎Moreover‎, ‎we can‎ ‎determine tensor squares of two-generator p-groups of class three‎ ‎satisfying the law $[x,g,y]\otimes g=1_{\otimes}$‎.


Main Subjects

D. P. Biddle and L-C. Kappe (2003). On subgroups related to the tensor center. Glasgow Math. J.. 45, 323-332 R. Brown, D. L. Johnson and E. F. Robertson (1987). Some computations of nonabelian tensor products of groups. J. Algebra. 111, 177-202 R. Brown and J. L. Loday (1984). Excision homotopique en basse dimension. C. R. Acad. Sci. Ser. I Math. Paris. 298, 353-356 R. Brown and J. L. Loday (1987). Van Kampen theorems for diagrams of spaces. Topology. 26, 311-335 G. J. Ellis (1995). Tensor products and q-crossed modules. J. London Math. Soc. (2). 51, 243-258 W. P. Kappe (2003). Some subgroups defined by identities. Illinois J. Math.. 47, 317-326 I. D. Macdonald (1962). On certain varieties of groups(II). Math. Z.. 78, 175-188 A. Magidin and R. F. Morse (2010). Certain homological functors of 2-generator p-groups of class 2, Computational Group Theory and the Theory of Groups II. Contemporary Mathematics. 511, 127-166 P. Moravec (2005). On nonabelian tensor analogues of 2-Engel conditions. Glasgow Math. J.. 47, 77-86 M. M. Nasrabadi and A. Gholamian (2012). On non abelian tensor analogues of 3-Engel groups. Indagationes Mathematicae. 23, 337-340 B. H. Neumann (1954). Groups covered by permutable subsets. J. London Math. Soc.. 29, 236-248 D. J. S. Robinson (1982). A course in the theory of groups. Springer-Verlag, New York. J. H. C. Whitehead (1950). A certain exact sequence. Ann. Math.. 51, 51-110