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
- [math]\displaystyle{ G = HK = \{ hk : h\in H, k\in K\} \text{ and } H\cap K = \{e\}. }[/math]
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
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.
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