GENERATING GROUPS BY CONJUGATION-INVARIANT SETS

Abstract

Let S be a generating set of a group G. We say that G has finite width relative to S if G = (S ∪ S-1)k for a suitable natural number k. We say that a group G is a group of finite C-width if G has finite width with respect to all conjugation-invariant generating sets of G. We give a number of examples of groups of finite C-width, and, in particular, we prove that the commutator subgroup F′ of Thompson's group F is a group of finite C-width. We also study the behavior of the class of all groups of finite C-width under some group-theoretic constructions; it is established, for instance, that this class is closed under formation of group extensions.