  In this paper we present a simple framework to study various distance
problems of permutations, including the transposition and block-interchange
distance of permutations as well as the reversal distance of signed
permutations. These problems are very important in the study of the evolution
of genomes. We give a general formulation for lower bounds of the transposition
and block-interchange distance from which the existing lower bounds obtained by
Bafna and Pevzner, and Christie can be easily derived. As to the reversal
distance of signed permutations, we translate it into a block-interchange
distance problem of permutations so that we obtain a new lower bound.
Furthermore, studying distance problems via our framework motivates several
interesting combinatorial problems related to product of permutations, some of
which are studied in this paper as well.
