a) Let G be a group. Deﬁne ∼ by the following: a ∼ b ⇐⇒ ∃ g ∈ G such that gag-1 = b.

Prove that ∼ is an equivalence relation.

(b) Suppose a ∈ Z(G). What elements are in the same cell as a with respect to the relation

∽?

(c) Let a ∈ G and deﬁne the centralizer of a, CG(a), as the subset

CG(a) = {g ∈ G : ga = ag}.

Prove that CG(a) ≤ G. If a ∈ Z(G) what can you conclude about CG(a)?

(d) (Bonus) Show that there is a bijective correspondence between elements equivalent (via

∼) to a ∈ G and left cosets of CG(a).

I understand a), I proved ~ is an equivalence relation. But I'm lost on the other three parts. This is not homework by the way per say, it was in my book and I'm studying for our final, this is one of the ones labeled "Very hard" basically. Thank you for your help

Just for clarity, Z(G) indicates the center of the group