  A number of fields, including the study of genome rearrangements and the
design of interconnection networks, deal with the connected problems of sorting
permutations in "as few moves as possible", using a given set of allowed
operations, or computing the number of moves the sorting process requires,
often referred to as the \emph{distance} of the permutation. These operations
often act on just one or two segments of the permutation, e.g. by reversing one
segment or exchanging two segments. The \emph{cycle graph} of the permutation
to sort is a fundamental tool in the theory of genome rearrangements, and has
proved useful in settling the complexity of many variants of the above
problems. In this paper, we present an algebraic reinterpretation of the cycle
graph of a permutation $\pi$ as an even permutation $\bar{\pi}$, and show how
to reformulate our sorting problems in terms of particular factorisations of
the latter permutation. Using our framework, we recover known results in a
simple and unified way, and obtain a new lower bound on the \emph{prefix
transposition distance} (where a \emph{prefix transposition} displaces the
initial segment of a permutation), which is shown to outperform previous
results. Moreover, we use our approach to improve the best known lower bound on
the \emph{prefix transposition diameter} from $2n/3$ to $\lfloor3n/4\rfloor$,
and investigate a few relations between some statistics on $\pi$ and
$\bar{\pi}$.
