Let \(G\) be a group. A subgroup \(N\leq G\) is normal, written \(N\normal G\), if \(gNg^{-1} = N\) for every \(g\in G\).
Let \(G\) be a group and \(N\leq G\). The following are equivalent:
\(N\normal G\).
\(gN = Ng\) for every \(g\in G\).
\(\psi(N) = N\) for every inner automorphism \(\psi \in \Inn(G)\).
\(N\) is the only subgroup of \(G\) conjugate to \(N\).