Podgrupa

Podgrupa, or subgroup, is a concept in group theory. If G is a group with an operation, a subset H ⊆ G is a podgrupa if H itself forms a group under the same operation. This requires that the identity element of G lies in H, H is closed under the group operation, and every element of H has its inverse also in H.

A standard criterion states that H ≠ ∅ and for all a, b ∈ H we have ab ∈ H and a^{-1} ∈ H. Equivalently, H ≠ ∅ and for all a, b ∈ H, ab^{-1} ∈ H. These conditions ensure that H inherits the group structure from G.

Examples illustrate the concept. In the additive group (Z, +), the set nZ = {nk : k ∈ Z} is a podgrupa for any fixed n ≥ 0. In the general linear group GL(n, F), the special linear group SL(n, F) = {A ∈ GL(n, F) : det(A) = 1} is a podgrupa. In the symmetric group S3, the set {e, (12)} is a podgrupa of order 2.

Key properties include that intersections of podgrupe are podgrupe, and the intersection of any family of podgrupe is a podgrupa. Generated subgroups, denoted ⟨S⟩ for a subset S ⊆ G, are the smallest podgrupa containing S. Each podgrupa H partitions G into left cosets gH, and the index [G : H] counts these cosets when finite.

A subgroup N is normal if gNg^{-1} = N for all g ∈ G; normal subgroups give rise to quotient groups G/N, whose elements are cosets of N. These concepts form the foundation of many constructions in algebra.