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 = HK = { hk : h ∈ H and k ∈ K } and H ∩ K = {e},

that is, if every element of G has a unique expression as a product hk where h in H and k in K. 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, a subgroup with a complement, in some sense, lets the group be factored into simpler pieces.

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.

References

• David S. Dummit & Richard M. Foote (2003). Abstract Algebra. Wiley. ISBN 978-0-471-43334-7.