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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.
Reviewer: S.Baskaran

MSC:
20D60 Arithmetic and combinatorial problems involving abstract finite groups
20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure