Equivalence for Transitive Group Actions

Suppose $H\le G$ (see Subgroup) and $G$ acts on the set $X$ of left cosets of $H$ in $G$, with corresponding Homomorphism (or Group Morphism) $\sigma :G\to S_X$. Then:

1. This action is always Transitive (Group Action)
2. The stabilizer of the identity coset is $H$.
3. The Kernel of $\sigma$ is the intersection of all conjugates of $H$. It is the largest Normal Subgroup of $G$ contained in $H$.