.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.