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Special elements in groups. (English) Zbl 0711.20014
Group theory, Proc. 2nd Int. Conf., Bressanone/Italy 1989, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 23, 33-42 (1990).
[For the entire collection see Zbl 0695.00013.]
An element a of a group G is called (n,m)-special if \(| K^ n| \leq m\) for every \(K=\{a,g\}\), where g is an arbitrary element of G. Let \(S_{n,m}(G)\) be the set of all (n,m)-special elements of G. It is proved that \(S_{2,3}(G)\) and \(S_{3,5}(G)\) are always normal subgroups of G. Obviously, \(Z(G)\subset S_{2,3}(G)\). All finite groups in which \(Z(G)\neq S_{2,3}(G)\) are described and some open problems raised.
Reviewer: B.M.Schein
20D60 Arithmetic and combinatorial problems involving abstract finite groups
20E07 Subgroup theorems; subgroup growth
20F12 Commutator calculus