Question

一道群论证明题，多谢帮助！急！！！！！！！！！！~~~~~~

Let G be a group. Let a ∈ G. An inner automorphism of G is a
function of the form γ_a : G → G given by γ_a(g) = aga^(−1).
Let Inn(G) be the set of all inner automorphisms of G.
(a) Prove that Inn(G) forms a group. (starting by identifying an appropriate binary operation.)
(b) Define ϕ : G → Inn(G) by ϕ(a) = γ_a. Verify that ϕ is surjective
homomorphism and identify the kernel of ϕ.

Answer 1

(a)a _a(y_b(g))=y_a(bgb^-1)=abgb^-1a^-1=y_ab(g),so y_ay_b=y_ab∈ lnn(G)
[y_a(g)]^-1=(aga^-1)^-1=a^-1ga=y_a^-1(g),so (y_a)^-1=y_a^-1∈ lnn(G)
then lnn(G) is a group
(b)f(a)=y_a,(a)have improved f(a)is a homorphism G → Inn(G)
obversly,f(a) is a surjective function ,thus,f is surjective homorphism
f(a)=aga^-1=f(1)=g,a belongs to the center of G,so the kernel is C(G)