\begin{definition}[Definition:Complement of Subgroup/Definition 1]
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 $H$ {{iff}}:
: $G = H K$ and $H \cap K = \set e$
\end{definition}