 
Summary: Complexity of Cayley Distance and other General Metrics on
Permutation Groups
1
Thaynara Arielly de Lima and 1,2
Mauricio AyalaRinc´on
Grupo de Teoria da Computa¸c~ao, Departamentos de 1
Matem´atica e 2
Ci^encia da Computa¸c~ao
Universidade de Bras´ilia, Bras´ilia D.F., Brazil
{thay@mat,ayala@}unb.br
Abstract. Permutation groups arise as important structures in group theory because many al
gebraic properties about them are wellknown, which makes modelling natural phenomena by
permutations of practical interest. This paper reviews the complexity of some problems involving
permutation groups. Usability of the involved algebraic notions is illustrated by problems such as
genome rearragement by reversals for which it is wellknown that for the case of unsigned and
signed sorting by reversals the time complexity is, respectively, NPhard and P. Reversal distance
is a particular metric and in this work more general metrics on permutation groups are considered
emphasizing on the Cayley distance. In particular, we point out an error in one of the polynomial
reductions applied in Pinch's approach atempting to proof that the subgroup distance problem for
Cayley distance is NPcomplete and following his approach we present a simplified and correct
