Finite coverings: A journey through groups, loops, rings and semigroups
Luise-Charlotte Kappe · Contemporary mathematics - American Mathematical Society · 2014
In 1975, Paul Erdös asked the question if there exists a finite bound on the cardinality of sets of pairwise noncommuting elements in a group provided every such set is finite. B.H. Neumann answered Erdös’ question in the affirmative by showing that every such group is central-by-finite and that the converse also holds. In an unpublished result R. Baer had shown earlier that a group is the union of finitely many abelian subgroups if and only if it is central-by-finite. All the proofs rely heavily on Neumann’s Lemma, stating that in a set of subgroups whose union is the whole group, all subgroups of infinite index can be removed and the union of the remaining subgroups is still the whole group. The question by Erdös, Neumann’s Lemma and finite coverings all make sense in other algebraic structures. The topic of this paper is a survey of analogues of the above results for groups in the case of loops, rings and semigroups. In addition some results for groups concerning finite coverings are mentioned because they appear to make interesting topics for investigation in other algebraic structures.