Small Systems of Generators of Groups

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.