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 
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".
Relation to other products.
Complements generalize both the direct product (where the subgroups "H" and "K" are normal in "G"), and the semidirect product (where one of "H" or "K" is normal in "G"). The product corresponding to a general complement is called the internal Zappa–Szép product. When "H" and "K" are nontrivial, complement subgroups factor a group into smaller pieces.
Existence.
As previously mentioned, complements need not exist.
A "p"-complement is a complement to a Sylow "p"-subgroup. Theorems of Frobenius and Thompson describe when a group has a normal "p"-complement. Philip Hall characterized finite soluble groups amongst finite groups as those with "p"-complements for every prime "p"; these "p"-complements are used to form what is called a Sylow system.
A Frobenius complement is a special type of complement in a Frobenius group.
A complemented group is one where every subgroup has a complement.