On S-length of groups - Macedońska, Olga; Potyka, Aleksandra (Annales Mathematicae Silesianae, 2008)
Let G be a group and S be a subsemigroup in G, generating G as a group. Every element in G is a product of elements from S∪S−1. An equality G = S−1S · · · S−1S allows to define an S-length l(G) of the group G. The note concerns the problem posed by J. Krempa on possible values of l(G). We show that for collapsing groups, supramenable groups and groups of a subexponential growth l(G) ≤ 2. The S-length of a relatively free group can be equal to 1 or 2 or infinity, but it can not be equal to 3. The problem concerning other values is open.