On infinite groups (Q2653664)

Miscellaneous remarks on infinite groups, of which we quote some typical examples: Poincaré's theorem on the intersection of subgroups of finite index. If a group is covered by a finite set of subgroups then the reciprocals of their indices add up to more than 1 [for a generalization of this, cf. \textit{B. H. Neumann}, J. Lond. Math. Soc. 29, 236--248 (1954; Zbl 0055.01604), Corollary (4. 5)]. A generalization of the notion of ``layer'' of elements of fixed order leads to the theorem that the subset of elements having infinite order modulo a subgroup \(H\) of \(G\) is empty or has the same cardinal as \(G\). Characterization of abelian groups with certain special properties.