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- Complement_(group_theory) abstract "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 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).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.".
- Complement_(group_theory) wikiPageID "11436087".
- Complement_(group_theory) wikiPageRevisionID "598723773".
- Complement_(group_theory) hasPhotoCollection Complement_(group_theory).
- Complement_(group_theory) subject Category:Group_theory.
- Complement_(group_theory) comment "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.".
- Complement_(group_theory) label "Complement (group theory)".
- Complement_(group_theory) label "Complément d'un sous-groupe".
- Complement_(group_theory) sameAs Complément_d'un_sous-groupe.
- Complement_(group_theory) sameAs m.02rcg6k.
- Complement_(group_theory) sameAs Q2990581.
- Complement_(group_theory) sameAs Q2990581.
- Complement_(group_theory) wasDerivedFrom Complement_(group_theory)?oldid=598723773.
- Complement_(group_theory) isPrimaryTopicOf Complement_(group_theory).