Example. Corner and edge stabilizers of the cube [frap-000Y]
Example. Corner and edge stabilizers of the cube [frap-000Y]
Let \(G\) be the actual permutation group generated by the six outer-face quarter-turns of the ordinary cube, with its frame of reference fixed. Use the corner and edge fibers: cyclic corner labels with local group \(C_3\), and edge labels with local group \(C_2\). This specializes the local-orientation construction of [marchetti2025rubik, Section 3.1]; the orbit argument below identifies the stabilizers needed for transfer.
The action on eight corner positions is transitive. Adjacent corners share a face whose quarter-turn takes one to the other, and the cube's edge graph is connected. Hence a corner stabilizer \(H_c\) has index eight. Its action on the three cyclically ordered labels gives a homomorphism \(\rho _c:H_c\to C_3\). The action on twelve edge positions is also transitive: the four edges of one face form a single orbit under its quarter-turn; neighboring faces share an edge; and the face-adjacency graph is connected. Thus an edge stabilizer \(H_e\) has index twelve, with local action \(\rho _e:H_e\to C_2\).
Choose a frame at the fixed corner or edge and transport it using coset representatives. The transfer comparison gives
\[\chi _c=(\rho _c)_{\mathrm {ab}}\circ T_{G,H_c},\qquad \chi _e=(\rho _e)_{\mathrm {ab}}\circ T_{G,H_e}.\]We identify the abelianizations of \(C_3\) and \(C_2\) with those groups. The face-turn calculation proves that the twist and flip characters vanish on every legal generator, hence on \(G\). It follows that these two local images of transfer vanish. It does not follow that \(T_{G,H_c}\) or \(T_{G,H_e}\) is trivial in the larger stabilizer abelianizations. No classification of \(G\) was needed.