normal subgroup

Let \(G\) be a group, \(U \subseteq G\) a subgroup. Then \(U\) is said to be a normal subgroup, \(U \trianglelefteq G\), if for \(g \in G\) and \(u \in U\) it holds, that

\begin{align*} g^{-1} \cdot u \cdot g \in U \end{align*}

Note that this condition is equivalent to

\begin{align*} g \cdot u \cdot g^{-1} \in U \end{align*}

as one may choose \(g = g'^{-1}\) and get

\begin{align*} g'^{-1} \cdot u \cdot g' \in U \end{align*}