Originally Posted by

**MichaelMath** Consider $\displaystyle S_5$, the symmetric group of degree five. Does it have a subgroup isomorphic to $\displaystyle C_5 \times C_5$ ?

Of course not: $\displaystyle |C_5\times C_5|=25\nmid 120=5!$ , so by Lagrange it can't be

Does it have elements of order $\displaystyle 6$?

Lots of: any element of the form $\displaystyle (ab)(cde)$ , with different a,b,c,d,e has order 6

Does it have a subgroup isomorphic to $\displaystyle D_5$ ?

Check the subgroup generated by $\displaystyle (14)(23),\,(12345)$

What about a subgroup isomorphic to $\displaystyle D_6 $?

Check the subgroup $\displaystyle <(34),\,(12)(345)>$

Let $\displaystyle C \subset S_5$ be a cyclic subgroup of order three. Find, up to isomorphism, the centraliser and the normaliser of $\displaystyle C$.

You try this yourself.

Tonio