Normal subgroup

A subgroup N of a group G is a normal subgroup (written $N\triangleleft G$) if one of the following equivalent conditions holds:

1. $aN=Na$ for all $a\in G$
2. $aNa^{-1}\subset N$ for all $a\in G$
3. $aNa^{-1}=N$ for all $a\in G$