Normal subgroup of a group (awaiting review)

If $H$ is a subgroup of a 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 binary operation on the set of left cosets of $H$

If $H$ is a normal subgroup, we write $H \lhd G$, and the set of left cosets $G/H$ form a group under the operation given in (4) above.