Normal subgroup and conjugate classes

Question

My solution I am not sure if my solution is correct .

Answer 1

Let $\rho$ be the congruence relation. We will denote $g_1\sim g_2$ to mean $(g_1,g_2)\in \rho$.

Define $H:= [e] = \{g\in G: g\sim e\}$, the equivalence class of the identity.

Your job is to show that $H$ is a normal subgroup, and that it induces $\rho$. Here is one step of that proof:

$H$ is closed under conjugacy because if $h\sim e$ then $gh \sim g$ and $ghg^{-1} \sim gg^{-1} = e$ by definition of a congruence relation.

So it remains to check that $H$ satisfies group axioms, and that $\rho = \{(a,b) \in G\times G |a^{-1}b \in H\}$.