Normal subgroup of a group

If $H$ is a {{KNOWL('group.subgroup', 'subgroup')}} of a {{KNOWL('group', 'group')}} $G$, then $H$ is **normal** if any of the following equivalent conditions hold: 1. $gHg^{-1}=H$ for all $g\in G$ 2. $gHg^{-1}\subseteq H$ for all $g\in G$ 3. $gH=Hg$ for all $g\in G$ 4. $(aH)*(bH)=(ab)H$ is a well-defined {{KNOWL('alg.binary_operation', 'binary operation')}} on the set of left cosets of $H$