EXERCISES Exercise 1. Find the center and all conjugacy classes in dihedral group D2", n 2 3. Exercise 2. Let G be a nite group, so that C/Z(G) is cyclic. Show that G must be abelianT so G = Z(G) and G/Z(G) is trivial. So, to rephrase this. if G is nonAbelian, then G/Z(G) is not. cyclic. Hint. Let G/Z(G} be cyclic. Let 93(0) be a generator of G/Z(G). This means that for any I E G its coset rZ(G) must equal to (gZ(G')}" = g\"Z(G') for some n E Z. Now conclude that there exists 2 E 27(0) so that 3: 2 97's. Now take y E C. Using exactly same argument, we see that there exists 2' E Z (G) so that In ,7 y = 9 J for some in E Z. Now. show that r, y connnute. I: Exercise 3. Let G be a prime number. and let G be a. group of order p2. Show that G must be abelian. Hint. Since G is a pgroup, by a result from the class we see that Z (G) is non trivial. If G is nonabelian, then by Lagrange, |Z(G)[ = p, but then |G/Z(G}| must be cyclic, which contradicts the previouis problem. I: Exercise 4. Let G be a nite group, J: E G. Show that number of elements in the conjugacy class of a: is a divisor of ICU |xf Hint. Recall the crucial formula: If G acts on a set X . J: E X then the number of elements in the orbit of 3: is (G : GI}, where G1 is the stabilizer of J: in G. Now, take X : G and the action is the conjugation. Then the stabilizer of a: is the centralizer C(33) : {g E G : girg'l : 1'}. Now, observe that. (as) is a subgroup of C(36). I: Exercise 5. Let H be a subgroup of a group G of index 2. so (G : H) : 2. Show that H must. be anormal subgroup of G. Hint. \"'0 know that number of distinct left and right cosets are the same. Thus there are exactly thwo left cosets H ,G \\ H. Also. there are two right cosets H. G \\ H. Now show that for any g E G we have 9h = Hg