.Consider , the symmetric group of degree five. Does it have a subgroup isomorphic to ?

Of course not: , so by Lagrange it can't be

Does it have elements of order ?

Lots of: any element of the form , with different a,b,c,d,e has order 6

Does it have a subgroup isomorphic to ?

Check the subgroup generated by

What about a subgroup isomorphic to ?

Check the subgroup

Let be a cyclic subgroup of order three. Find, up to isomorphism, the centraliser and the normaliser of .

You try this yourself.

Tonio