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Zbl 0754.20010
Lennox, John C.; Hassanabadi, A.Mohammadi; Wiegold, James
Some commutativity criteria. II. (English)
[J] Rend. Semin. Mat. Univ. Padova 86, 207-211 (1991). ISSN 0041-8994

This paper follows the authors' earlier part I [ibid. 84, 135-141 (1990; Zbl 0725.20025)]. They call a group a $P\sb n$-group if all sets of $n$ elements commute, and the main result of the earlier paper was that all infinite $P\sb n$-groups are abelian. They now turn their attention to finite $P\sb n$-groups, and show that such a group is necessarily abelian if its order is at least $2n$, but that all groups of order less than $2n$ are $P\sb n$-groups (Theorem 1). By contrast they show that monoids (that is: semigroups with a neutral element), for which the definition clearly also makes sense, behave rather differently, by determining (Theorem 2) the exact structure of all non-commutative $P\sb 2$-monoids. They mention the unsolved problems of the structure of non-commutative $P\sb 3$-monoids and non-commutative $P\sb 2$-semigroups without a neutral element.
[B.H.Neumann (Canberra)]

MSC 2000:: *20F05 Presentations of groups
20E34 General structure theorems of groups
20D99 Abstract finite groups
20M05 Free semigroups
20K99 Abelian groups
20F99 Special aspects of infinite or finite groups

Keywords: finite $P\sb n$-groups; non-commutative $P\sb 2$-monoids; non-commutative $P\sb 3$-monoids; non-commutative $P\sb 2$-semigroups

Citations: Zbl 0725.20025

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