We can think of as with the relation .
And here is the subgroup diagram.
1. Find all the normal subgroups of <a> of S3 and <b> of D4, where a is in S3 and b is in D4. I know A4 is a normal subgroup of S3. Are there any others?
2. Let H be a subgroup of an arbitrary group G. Prove H is normal iff it has the following property: For all a,b, in G, ab is in H iff ba is in H.