Equivalence of Definitions of Complement of Subgroup

Theorem

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

Let $H$ and $K$ be subgroups.

The following definitions of the concept of Complement of Subgroup are equivalent:

Definition $1$

$K$ is a complement of $H$ if and only if:

$G = H K$ and $H \cap K = \set e$

Definition $2$

$K$ is a complement of $H$ if and only if:

$G = K H$ and $H \cap K = \set e$

Proof

Definition $1$ implies Definition $2$

Let $G = H K$.

Then $H K$ is a group.

By Subset Product of Subgroups:

$H K = K H$

Thus $K H = G$.

$\Box$

Definition $2$ implies Definition $1$

Let $G = K H$.

Then $K H$ is a group.

By Subset Product of Subgroups:

$H K = K H$

Thus $H K = G$.

$\blacksquare$