Let be the group of bijections from to itself which are continuous outside a finite set. Let be its quotient by the subgroup of finitely supported permutations. We show that the Kapoudjian class of vanishes. That is, the quotient map splits modulo the alternating subgroup of even permutations. This is shown by constructing a nonzero group homomorphism, called signature, from to . Then we use this signature to list normal subgroups of every subgroup of which contains such that , the projection of in , is simple.
Cite this article
Octave Lacourte, Signature for piecewise continuous groups. Groups Geom. Dyn. 16 (2022), no. 1, pp. 75–84DOI 10.4171/GGD/664