Definition:Complement of Subgroup/Definition 1

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

Let $H$ and $K$ be subgroups.

Let $HK$ be their subset product and $H \cap K$ their intersection.

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

Also see

 * Equivalence of Definitions of Complement of Subgroup