Show that 8(x) E Q, which will h.q.h-¹ EhQh-¹ is also an element of Q. Hence 8(x) E Q so that x E P. Therefore , so that (x) = h-q-h-1. Since Q is normal in H, we have hQh-¹C Q. Thus Hand q = 0(p) E Q. Then, 8(x) = 0(gpg-¹) = 0(g)-0(p)-0(g)-¹ = gPg¹ SP and P is normal in G, as desired. Exercises When working with the group D4, refer to Appendix B for its group table. 1. Let H = {e, d} be a subgroup of D4- (a) Give an example which shows that the coset multiplication shortcut does not hold in D4/H. (b) Give an example which shows H is not normal in D4. 2. Let H = {e, d}, K = {E, U}, Z = {e, r180) be subgroups of D4. Note: By Exercise #1(b) and Example 24.14, respectively, H and K are not normal in D4. (a) Explain why Z is normal in D4. (b) Compute the set product HK = {aß | a EH, BEK). Is HK a subgroup of D4? Explain. om (c) Compute the set product ZK, defined similarly. Is ZK a subgroup of D₂? Ex- implain. a 3. Determine if each statement is true or false. If it's true, prove it. If it's false, give a counterexample. (a) Let H be a normal subgroup of G. If H and G/H are commutative, then G is commutative. (b) If H is a normal subgroup of G and K is a normal subgroup of H, then K is normal in G. 4. Let N be a normal subgroup of G and let H be a subgroup of G (but not necessarily normal in G). Define the set product NH = {nh | nEN, h E H}. Prove that NH is a subgroup of G. 5. Consider the subgroup K = {e, r90, 180, 1270) of D4. In this problem, we'll analyze why the set inclusion dK UK C (dv)K must hold. Note: Your work from Chapter 19, Exercise #16 will come in handy here. . (a) Find the element? EK such that roov = v? (b) Let aß e dK UK, where a = droo E dK and ß = vr180 € UK. Using only your result from part (a), explain why aß € (dv)K. (c) Explain why dKvK C (dv)K.