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.



Smallest subgroups with trivial centralizer?
The number $k(G)$ is the domination number of the non-commuting graph of $G$. 
See Proposition 2.14 of [J. Algebra, 298 (2006) 468–492].


By Corollary 2.17 of [J. Algebra, 298 (2006) 468–492], if $k(H)$ is finite for some finite index subgroup $H$ of $G$, then $k(G)$ is also finite. 


Another meaning for $k(G)$ is the minimum number of elements defining the inner automorphism group of the group $G$ (with trivial center). You may find it in the following paper:


Giovanni Cutolo and Chiara Nicotera, Subgroups defining automorphisms in locally nilpotent groups, Forum Mathematicum, Volume 15, Issue 4, (2003) 489-506. 


Donwload the latter paper from


<http://www.dma.unina.it/cutolo/papers/Forum03.pdf>


