Remark. When a character is conserved [frap-000T]
Remark. When a character is conserved [frap-000T]
For finite \(I\), the ambient coordinate group \(W=K\wr \operatorname {Sym}(I)\) has the total-orientation character \(\chi (\pi ,a)=\prod _i[a_i]\in K_{\mathrm {ab}}\). If a puzzle's allowed moves form a set \(S\subseteq W\), its legal group is \(G=\langle S\rangle \), usually a proper subgroup of \(W\). Then
\[G\subseteq \ker \chi \quad \Longleftrightarrow \quad \chi (s)=1\quad \hbox {for every }s\in S.\]The forward implication includes each generator. Conversely the kernel of a homomorphism is a subgroup, so it contains all products and inverses of generators lying in it. This is the generator-check step discussed after the canonical character in [marchetti2025rubik, Section 3.1, p. 7]. Its premise must be verified for the chosen moves.
The nonabelian product of local factors need not be independent of order or frames. Abelianization makes reindexing and frame cancellation valid, but neither of those operations says that \(\chi (s)=1\). The cycle criterion supplies a sufficient test, applied explicitly to cube face turns.