Definition:Complement of Subgroup

Definition
Let $G$ be a group with identity $e$.

Let $H$ and $K$ be subgroups.

Definition 1
$K$ is a complement of $H$ :
 * $G=HK$ and $H\cap K = \{e\}$

Definition 2
$K$ is a complement of $H$ :
 * $G=KH$ and $H\cap K = \{e\}$

Also see

 * Equivalence of Definitions of Complement of Subgroup