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

G

=

H

K

=

{

h

k

:

h

∈

H

,

k

∈

K

}

and

H

∩

K

=

{

e

}

.

{\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.

Properties

Complements need not exist, and if they do they need not be unique. That is, H could have two distinct complements K1 and K2 in G.

If there are several complements of a normal subgroup, then they are necessarily isomorphic to each other and to the quotient group.

If K is a complement of H in G then K forms both a left and right transversal of H. That is, the elements of K form a complete set of representatives of both the left and right cosets of H.

The Schur–Zassenhaus theorem guarantees the existence of complements of normal Hall subgroups of finite groups.

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.