Math 113, Homework 7 Due 10/24/16 at 10:10 am 1. What are the conjugacy classes of D4 ? (\"Conjugacy class\" is a shorter way of saying \"orbit under the action of conjugation\".) Which subgroups of D4 are unions of conjugacy classes? 2. The previous exercise may have suggested a conjecture: A subgroup is normal if and only if it is a union of conjugacy classes. Prove this. (We didn't prove Theorem 14.13 in class, but feel free to use it.) 3. Suppose that G contains normal subgroups H and K with H K = {e} and HK = G, where by HK we mean {hk|h H, k K}. (a) Show that if h H and k K then hk = kh. (b) Show that each element of G can be written uniquely as an element of H times an element of K. (c) Show that G is isomorphic to H K. 4. Strengthen our result from class by showing that any group of size p2 (p a prime) must be isomorphic to Zp2 or Zp Zp 5. Show that any group, G, of size pq (where p and q are prime) is either isomorphic to Zp Zq or else Z(G) = 1. 6. Prove the following theorem: if each element of a group, G, has order which is a power of p, then the order of G is also a prime power. We skipped the classification of finite abelian groups, but for the purposes of this problem, assume you know that the theorem is true for abelian groups (Hint: use a proof by induction, and in the class formula remember that the size of the orbit of x is |G|/|stab(x)|.) 7. It is easy to think that E is transitive (i.e if H E K E G then H E G.) Think about D4 until you come up with a counterexample. 1