Signature for piecewise continuous groups (Q2141702)

Let \(X\) denote right-open, left-closed interval \([0,1)\). Group of bijections from \(X\) to \(X\) is denoted by \(\mathfrak{S}(X)\), and the set of all permutations of \(X\) continuous outside a finite set is a subgroup denoted by \(\widehat{PC^{\bowtie}}\). Let \(PC^{\bowtie}\) be the quotient of \(\widehat{PC^{\bowtie}}\) by the subgroup of finitely supported permutations, denoted by \(\mathfrak{S}_{\mathrm{fin}}\). The author proves that the Kapoudjian class of \(PC^{\bowtie}\) vanishes. This is done through the following main results of the paper: Theorem 1.1 There exists a group homomorphism \(\varepsilon:\widehat{PC^{\bowtie}}\rightarrow\mathbb{Z}/2\mathbb{Z}\) that extends the classical signature on \(\mathfrak{S}_{\mathrm{fin}}\). Corollary 1.2 Let \(G\) be a subgroup of \(PC^{\bowtie}\). Then the Kapoudjian class of \(G\) is zero. Later, in section 4, author applies these results to study the normal subgroups of \(\widehat{PC^{\bowtie}}\), and other certain subgroups. More specifically, it is shown that if \(\widehat{G}\) is a subgroup of \(\widehat{PC^{\bowtie}}\) containing \(\mathfrak{S}_{\mathrm{fin}}\), such that its projection in \(\widehat{PC^{\bowtie}}\) is simple nonabelian, then \(\widehat{G}\) has exactly five normal subgroups.