A subgroup \(N\) of a group \(G\) is called normal and denoted as \(N\unlhd G\), if and only if:
(1) its left and right cosets with respect to any element \(a\in G\) coincide, i.e. \[aN=Na,~\forall a\in G.\]
(2) for all \(a\in G\): \[aNa^{-1}=N\]
where the set \(aNa^{-1}\) is defined as \(\{g\in G~|~g=a\ast h\ast a^{-1},~h\in N\}.\)