Paper 3, Section II, D

Let $S_{n}$ be the group of permutations of $\{1, \ldots, n\}$, and suppose $n$ is even, $n \geqslant 4$.

Let $g=(12) \in S_{n}$, and $h=(12)(34) \ldots(n-1 n) \in S_{n}$.

(i) Compute the centraliser of $g$, and the orders of the centraliser of $g$ and of the centraliser of $h$.

(ii) Now let $n=6$. Let $G$ be the group of all symmetries of the cube, and $X$ the set of faces of the cube. Show that the action of $G$ on $X$ makes $G$ isomorphic to the centraliser of $h$ in $S_{6}$. [Hint: Show that $-1 \in G$ permutes the faces of the cube according to $h$.]

Show that $G$ is also isomorphic to the centraliser of $g$ in $S_{6}$.

*Typos? Please submit corrections to this page on GitHub.*