Small Systems of Generators of Groups

Abstract

A subset S of a group G is said to be large (left large) if there is a finite subset K such that G=KS=SK (G=KS). A subset S of a group G is said to be small (left small) if the subset G\setminus KSK (G\setminus KS) is large (left large). The following assertions are proved: (1) every infinite group is generated by some small subset; (2) in any infinite group G there is a left small subset S such that G=SS -1; (3) any infinite group can be decomposed into countably many left small subsets each generating the group.