  Given a generator set $S$ of the symmetric group ${\rm{Sym}}_n$, every
permutation $\pi\in {\rm{Sym}_n}$ is a word (product of elements) of $S$. A
positive integer $d(\pi)$ is associated with each $\pi\in{\rm{Sym}_n}$ taking
the length of the shortest such word, and the $S$-diameter $d(S)$ is the
maximum value of $d(\pi)$ with $\pi$ ranging over ${\rm{Sym_n}}$. The distance
$d(\pi,\nu)$ of two permutations $\pi,\nu$ defined by $d(\nu^{-1}\circ\pi)$
satisfies the axioms of a metric space. In this paper we consider the case
where $S$ consists of all block transpositions of ${\rm{Sym_n}}$ and call
$d(\pi)$ the block transposition distance of $\pi$. A strong motivation for the
study of this special case comes from investigations of large-scale mutations
of genome, where determining $d(\pi)$ is known as sorting the permutation $\pi$
by block transpositions. In the papers on this subject, toric equivalence
classes often play a crucial role since $d(\pi)=d(\nu)$ when $\pi$ and $\nu$
are torically equivalent. A proof of this result can be found in the
(unpublished) Hausen's Ph.D Dissertation thesis; see \cite{Ha}. Our main
contribution is to obtain a bijective map on ${\rm{Sym}_n}$ from the toric
equivalence that leaves the distances invariant. Using the properties of this
map, we give an alternative proof of Hausen's result which actually fills a gap
in the proof of the upper bound on $d(S)$ due to Eriksson and his coworkers;
see \cite{EE}. We also revisit the proof of the key lemma \cite[Lemma 5,1]{EE},
giving more details and filling some gaps.
