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.
Complements generalize both the direct product (where the subgroups H and K commute element-wise), and the semidirect product (where one of H or K normalizes the other). The product corresponding to a general complement is called the Zappa–Szép product. In all cases, complement subgroups factor a group into smaller pieces.
Some properties of complement subgroups:
- 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.
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
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.