# Rank of Conjugate Closure of a Subset

How does one find the rank of a conjugate closure of a subset?

In particular, I am studying this group: Let $K_n$ be the group with $n$ generators $x_1,\cdots,x_n$ satisfying the relation that each $x_i$ commutes with all its conjugates, where $i=1,\cdots,n$. Thus there is a group homomorphism $f_{n}:K_n\to K_{n-1}$ mapping $x_i$ to $x_i$ when $i\leq n$ and maps $x_n$ to the identity element. Consider the kernel of $f$: it is the conjugate closure of the one-element subset $\{x_n\}$ of $K_n$. Then what is the rank (I mean the minimum number of generators) of the conjugate closure of $\{x_n\}$?

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Is conjugate closure the same as normal closure? – Nick Gill Jul 4 '13 at 8:58
Yes, they are exactly the same. – Zuriel Jul 4 '13 at 11:16
As stated the rank of the normal closure of $\{ x_n \}$ in $K_n$ could be 1 if $K_n$ is abelian and infinite if $K_n$ is the free group. Do you also mean that $x_i$ only commutes with elements it is conjugate to? – Neil Hoffman Jul 10 '13 at 20:47
@NeilHoffman, Yes, $K_n$ is neither Abelian (unless $n=1$) nor the free group. In $K_n$, each generator $x_i$ commutes with all the conjugates of $x_n$. When $n=2$, it is the discrete Heisenberg group. – Zuriel Jul 15 '13 at 4:18