Small systems of generators of groups. (English. Russian original) Zbl 1078.20032

Protasov, I. V.

Math. Notes 76, No. 3, 389-394 (2004); translation from Mat. Zametki 76, No. 3, 420-426 (2004).

Summary: 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.

MSC:

20F05

Generators, relations, and presentations of groups

Keywords:

infinite groups; left large subsets; left small subsets; generators