Definition:Complement of Subgroup
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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
- Results about subgroup complements can be found here.