Let $G$ be a countable group with trivial center. Define $k(G)$ to be the smallest value of $k$ such that there is a subgroup $H$ of $G$ generated by $k$ elements having trivial centralizer in $G$, and $k(G)=\infty$ if no such subgroup exists. Is there a name for this invariant of $G$? Have its properties been studied?

My motivation comes from working on the automorphism group of a shift of finite type (see *The Automorphism Group of a Shift of Finite Type*, Trans. AMS **306** (1988), 71-114 by Boyle, me, and Rudolph), where $k(G)$ may give some insight into unsolved problems such as whether the automorphism groups of the 2-shift and 3-shift are isomorphic as groups.

How to Write Mathematicshe talks about "frozen notation," asking "Who would dare write 'Let 6 be a group'?"! $\endgroup$