Definition:Complement of Subgroup/Definition 1

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

Let $H$ and $K$ be subgroups of $G$.

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

$K$ is a complement of $H$ :
 * $G = H K$ and $H \cap K = \set e$

Also see

 * Equivalence of Definitions of Complement of Subgroup