Category:Definitions/Subgroup Complements

This category contains definitions related to Subgroup Complements.
Related results can be found in Category:Subgroup Complements.

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 $H \cap K = \set e$