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.

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. 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.

See also

• Product of group subsets

References

• David S. Dummit & Richard M. Foote (2003). Abstract Algebra. Wiley. ISBN 978-0-471-43334-7.
• I. Martin Isaacs (2008). Finite Group Theory. American Mathematical Society. ISBN 978-0-8218-4344-4.