Invariant Subgroups
Definition 1.11 – Invariant Subgroup
An invariant subgroup is the union of equivalence classes within a group.
For example, the $D_{3}$ group has three equivalence classes: $\cal{C}_{1} = \{ e \}$, $\cal{C}_{2} = \{ c_{1}, c_{2} \}$, $\cal{C}_{3} = \{ b_{1}, b_{2}, b_{3} \}$. The subgroups are $\{ e, c_{1}, c_{2} \}$, $\{ e, b_{1} \}$, $\{ e, b_{2} \}$ and $\{ e, b_{3} \}$. Of the subgroups, only $\{ e, c_{1}, c_{2} \} = \cal{C}_{1} \cup \cal{C}_{2}$. Therefore, $\{ e, c_{1}, c_{2}\}$ is an invariant subgroup.
We denote an invariant subgroup $H$ of group $G$ : $H \triangleleft G$.
Theorem 1.6
The right and left cosets of an invariant subgroup $H$ of group $G$ are identical. In other word,
$$ \text{If } H \triangleleft G, \space Hg = gH \space \forall g \in G $$Proof
Consider $H \triangleleft G$, where $H = \{ h_{1}, h_{2}, \dots, h_{m} \}$.
$$ \begin{align} Hg &= \{h_{1}g, h_{2}g, \dots, h_{m}g\} \\ &= \{ gh_{1}g^{-1}g, gh_{1}g^{-1}g, \dots, gh_{m}g^{-1}g \} \qquad \text{using Theorems 1.3 and 1.5} \\ &= g\{ h_{1}, h_{2}, \dots, h_{m} \} = gH \end{align} $$