Prove that a subgroup $N$ of a group $G$ is normal iff $ab \in N \Leftrightarrow ba \in N . \forall a,b \in G $

Question

Prove that a subgroup $N$ of a group $G$ is normal iff $ab \in N \Leftrightarrow ba \in N . \forall a,b \in G $

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$\Leftarrow$ $ab \in N \iff ba \in N , \forall a,b \in G$ then N is a normal

following the hint $a^{-1} n=b$ not sure but

$$n_1 g =(ab)g =(a a^{-1}n)g=n_1 g$$

$\Rightarrow $ $N$ is normal then $ab \in N \iff ba \in N , \forall a,b \in G$

so $$gn_1=n_2 g$$

guessing that $n_1 =ab \in N$ not sure here let $n_1 =ab??? $ lost so $$ g(ab)=n_2 g$$

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Answer 1

If $N$ satisfies $\forall a,b \in G ab \in N \Leftrightarrow ba \in N$ Then if $N \ni a =b(b^{-1}a) \Rightarrow b^{-1}ab \in N$, so N is normal.

If $N$ is normal, then $ab \in N \Rightarrow N \ni b(ab)b^{-1} = ba$.