a) Let G and H be groups and let 7 : G H be an isomorphism....

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a) Let G and H be groups and let 7 : G H be an isomorphism. Show that (g-¹) = T: T(g)-¹ for any g € G. b) Let G be a group and a € G. Define C(a) = {g €G: ga = ag}. The set C(a) is called the centralizer of a. Show that C(a) is a subgroup of G, and find the centralizer of o in D3. [10] a) Let G and H be groups and let 7 : G H be an isomorphism. Show that (g-¹) = T: T(g)-¹ for any g € G. b) Let G be a group and a € G. Define C(a) = {g €G: ga = ag}. The set C(a) is called the centralizer of a. Show that C(a) is a subgroup of G, and find the centralizer of o in D3. [10]