Complement (group theory)

In mathematics, especially in the area of algebra known as group theory, a complement of a subgroup H in a group G is a subgroup K of G such that

${\displaystyle G=HK=\{hk:h\in H,k\in K\}{\text{ and }}H\cap K=\{e\}.}$

Equivalently, every element of G has a unique expression as a product hk where h ∈ H and k ∈ K. This relation is symmetrical: if K is a complement of H, then H is a complement of K. Neither H nor K need be a normal subgroup of G.