Definition:Complement of Subgroup

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.

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 $H \cap K = \set e$

Also known as

If $H$ is a complement of $K$ (and thus equivalently, if $K$ is a complement of $H$) the subgroups are said to be complementary.

Also see

• Equivalence of Definitions of Complement of Subgroup

• Results about subgroup complements can be found here.