1. | S₄ | = 24 Thus the only possible values for the orders of the subgroups are: 1, 2, 3, 4, 6, 8, 12, 24. Find a subgroup for each of these or prove no such subgroup exists. Can you find pairs of groups of the same order which are non-isomorphic? Please do this problem without using any outside references. Everything you need to do this problem has been covered in class. However DO feel free to form a group of 2-4 people to work on this together.

2. Can the product of two 3-cycles be a 2-cycle? Why or why not? Try doing this one yourself.

3. Can the product of two 2-cycles be a 3-cycle? Why or why not? Try doing this one yourself.

4. If g is a k-cycle, what can you say about gⁱ, i=1,2,...,k ? This is a strange question in that you’re not expected to make observations about all the possible values of i. However try to say something about SOME of the values of i. This is another good problem to work on in a small group.

5. Show that isomorphism is an equivalence relation. Another good problem to try on your own.

6. Let H₁ and H₂ be subgroups of G. And let H₁H₂ be the set of all elements of G of the form h₁h₂ where h₁ is in H₁ and h₂ is in H₂ . Find examples where H₁H₂ is a subgroup and examples where it is not a subgroup. Another good problem for group work.

7. Suppose G and G’ are isomorphic groups and f:G→G’ is an isomorphism. And suppose H≤G. Let H’=f(H). (I.e. H’ is the set of all f(h) for h in H.) Prove that H’ is a group.