Signature for piecewise continuous groups

Let PC be the group of bijections from [0, 1[ to itself which are continuous outside a finite set. Let PC be its quotient by the subgroup of finitely supported permutations. We show that the Kapoudjian class of PC vanishes. That is, the quotient map PC $rightarrow$ PC splits modulo the alternating subgroup of even permutations. This is shown by constructing a nonzero group homomorphism, called signature, from PC to Z 2Z. Then we use this signature to list normal subgroups of every subgroup G of PC which contains S fin and such that G, the projection of G in PC , is simple.