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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*.

- Complements need not exist, and if they do they need not be unique. That is,
*H*could have two distinct complements*K*_{1}and*K*_{2}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.

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.

As previously mentioned, complements need not exist.

A ** p-complement** is a complement to a Sylow

A **Frobenius complement** is a special type of complement in a Frobenius group.

A complemented group is one where every subgroup has a complement.

- 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.