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On two-element subsets in groups. (English) Zbl 0579.20017
Collective phenomena, 4th Int. Conf., Moscow 1981, Ann. N.Y. Acad. Sci. 373, 183-190 (1981).
[For the entire collection see Zbl 0568.00003.]
Let $$E=\{a,b\}$$ be a two-element subset of a group G. The second and third stage multiplication tables for E are, respectively, with the understanding that if $$ba=ab$$ the column of ba is not written and, similarly, if $$b^ 2=a^ 2$$ the column of $$b^ 2$$ is not written and thus the number of distinct columns of the third stage multiplication table is the number of distinct elements of the set $$\{a^ 2,ab,ba,b^ 2\}$$. Two two-element subsets E and F of a group G are said to be isomorphic up to the third stage if the second and third stage multiplication tables of F may be obtained from those of E by a proper change of rows and columns with lexicographical redesignations. The authors describe, up to isomorphism up to the third stage, all the two- element subsets existing in groups. They also discuss some results which illustrate some research directions arising from the notion of isomorphic sets in a group.
 20D60 Arithmetic and combinatorial problems involving abstract finite groups 20D20 Sylow subgroups, Sylow properties, $$\pi$$-groups, $$\pi$$-structure