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 H and K together generate G and the intersection of H and K is the identity. Equivalently, a subgroup K of G is called a complement for H in G if G = HK = { hk : h ∈ H & k ∈ K } and H ∩ K = {e}. 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 allows the group to be factored into simpler pieces.