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.
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Protasov, I.V. Small Systems of Generators of Groups. Mathematical Notes 76, 389–394 (2004). https://doi.org/10.1023/B:MATN.0000043466.45064.97
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DOI: https://doi.org/10.1023/B:MATN.0000043466.45064.97