Definition:Complement of Subgroup/Definition 1
Jump to navigation
Jump to search
Definition
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 $H$ if and only if:
- $G = H K$ and $H \cap K = \set e$
Also see
Sources
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $8$: Homomorphisms, Normal Subgroups and Quotient Groups: Exercise $22$
- 2003: David S. Dummit and Richard M. Foote: Abstract Algebra (3rd ed.) $\S 5.5$: Semidirect Products