# Smallest subgroups with trivial centralizer?

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.

• in the notation $k(G)$, usually $k$ is not a variable... but as you like!
– YCor
Jan 13, 2017 at 16:18
• Yes that can mean the group ring, but I don't think there's any danger here of confusion! Jan 13, 2017 at 16:58
• I guess there is a long list of such invariants (the smallest cardinal of a subset whose centralizer has some given smallness property etc); in a linear group one can consider the smallest cardinal of a subset generating Zariski-dense subgroup (e.g. I guess it's 2 whenever the group is not virtually solvable, and hence "$k(G)$" (I don't get used sorry, it sounds to my brain like saying "let $\mathbb{Z}$ be the field of complex numbers") would equal 2 if I'm correct. Another related object is the length of chains of centralizers (one can consider the sup, or the infimum among saturated chains).
– YCor
Jan 13, 2017 at 18:57
• In Halmos's essay How to Write Mathematics he talks about "frozen notation," asking "Who would dare write 'Let 6 be a group'?"! Jan 14, 2017 at 1:55

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: