Normal subgroup

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

1. <math>a N = N a </math> for all <math> a \in G</math>
2. <math>a N a^{-1} \subset N </math> for all <math> a \in G</math>
3. <math>a N a^{-1} = N </math> for all <math> a \in G</math>