Consider $\displaystyle S_5$, the symmetric group of degree five. Does it have a subgroup isomorphic to $\displaystyle C_5 \times C_5$ ? Does it have elements of order $\displaystyle 6$? Does it have a subgroup isomorphic to $\displaystyle D_5$ ? What about a subgroup isomorphic to $\displaystyle D_6 $? Let $\displaystyle C \subset S_5$ be a cyclic subgroup of order three. Find, up to isomorphism, the centraliser and the normaliser of $\displaystyle C$.