Past exam of the mathematics course of the University of Cambridge / 2014 / ia / Paper 3 / 5D (Groups)

Conjugation in the symmetric group sends $(12)$ to $(\sigma (1)\sigma (2))$. Hence $\sigma$ commutes with $g$ exactly when it preserves the unordered pair $\{1,2\}$. It can independently swap that pair and arbitrarily permute the remaining points. Therefore the centraliser of a transposition isLet $m=n/2$. The orbits of a group action generated by $h$ are its $m$ two-point blocks. Any centraliser element must permute those blocks, and can choose independently whether to swap the two points in each image block. Conversely every such choice commutes with $h$, since it preserves the partner relation. This centraliser of a fixed-point-free involution is the permutation wreath product $C_2\wr S_m$, with

Place the cube at the origin with face normals $\pm e_1,\pm e_2,\pm e_3$, and label opposite faces by $(1,2),(3,4),(5,6)$. Every cube symmetry is a signed permutation matrix: there are $2^{3}3!=48$ choices. Its group action on faces is faithful because fixing all faces fixes all their normal directions. Central inversion $-I$ exchanges every opposite pair and therefore acts as $h$. Since $-I$ commutes with every linear cube symmetry, the face-action image lies in $C_{S_6}(h)$. The preceding count gives $|C_{S_6}(h)|=48$, so the faithful face action is onto this centraliser:For the remaining isomorphism, let $R$ be the rotational symmetry group of a cube. Half the signed permutation matrices have determinant $1$, so $|R|=24$. Central inversion has determinant $-1$, and every orientation-reversing symmetry is $(-I)r$ for a unique $r\in R$. Since $-I$ is central and $R\cap \{I,-I\}=\{I\}$, multiplication gives a direct product of groups:The group action of $R$ on the four body diagonals is faithful. To prove this, choose direction vectors $(1,1,1),(1,-1,-1),(-1,1,-1),(-1,-1,1)$. They span ${\mathbb{R}}^3$, and their only linear relation is that their sum is zero. A rotation fixing all four diagonal lines sends each listed vector to itself or its negative. Applying the relation forces all four signs to agree. The all-negative choice is $-I$, which is not in $R$, so the rotation is the identity. Thus $R$ embeds in $S_4$; both have order $24$, giving $R\cong S_4$. Finally $C_{S_6}(g)\cong C_2\times S_4$ from part (i). ConsequentlyThe direct-product decomposition of the cube symmetry group supplies the second isomorphism, while the first comes from the specified face action.