GT

Question 1

properties of “conjugation by a”:
γ(a) : G->G

Answer 1

• γ(a) is an isomorphism
• γ(a) γ(b) = γ(ab)
• inverse of γ(a) is γ(a^(-1))
• H<=G ⟹ γ(a)(H) = aHa^(-1)<=G, H≅γ(a)(H)

Question 2

G abelian ⟹

Question 3

conjugacy class properties

Answer 3

• x conjugate to itself
• x∈C↓(y) ⟹ y∈C↓(x)
• x∈C↓(y) and y∈C↓(z) ⟹ x∈C↓(z)
• every group is a disjoint union of conjugacy classes