A variety of applications of a theorem of B. H. Neumann on groups (Q1376667)

\textit{B. H. Neumann} [J. Lond. Math. Soc. 29, 236-248 (1954; Zbl 0055.01604), Publ. Math. 3, 227-242 (1954; Zbl 0057.25603)]\ proved the following theorem: Let \(G\) be a group, and \(G_1,\dots,G_r\) be subgroups of \(G\). If \(G\) is a set union of a finite number of cosets of the \(G_i\) irredundantly then \(G_1\cap\dots\cap G_r\) is of finite index in \(G\). In this note our main purpose is to point out the utility of this theorem in a variety of subjects like Banach spaces, curves, division rings, projective geometry, Riemann surfaces and vector spaces.