Normality of the torsion part of a group given it is a subgroup

Question

Let $t(G)$ denote the set of torsion elements of a group $G$. Prove that if $t(G)$ is a subgroup of G, then it is a normal subgroup of $G$.

Answer 1

Let $a\in t(G)$ then $\forall b \in G$ I show that $a^b \in t(G)$. Since $t(G)$ is a subgroup this makes it normal.

Let $n$ be the order of $a$, since conjugation is an automorphism: $1=(a^n)^b=(a^b)^n$. So, the order of $a^b$ divides $n$ therefore it is finite.