Complement (group theory)

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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

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 hH and kK. 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.

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