podgrupe

Podgrupe, in the context of group theory, are subgroups of a group G. If G = (G, ⋅) is a group, a subset H ⊆ G is a podgrupa (subgroup) if (H, ⋅) is itself a group under the same operation. This requires that the identity element e of G belongs to H, that every element h ∈ H has its inverse h^{-1} also in H, and that H is closed under the operation: for all h1, h2 ∈ H, h1 ⋅ h2 ∈ H.

A practical subgroup test states that H is a subgroup of G if it is nonempty, contains the identity, and is closed under the group operation and taking inverses. Equivalently, for all a, b ∈ H, the product a ⋅ b^{-1} lies in H. If these conditions hold, H ≤ G is a subgroup. Subgroups can be used to study the structure of G by examining how it decomposes into smaller, simpler pieces.

Examples help illustrate the concept. In the group of integers under addition, 2Z (the even integers) is a subgroup of Z. In the multiplicative group of nonzero real numbers, the set {1, −1} is a subgroup. For any element g in G, the cyclic subgroup generated by g, denoted ⟨g⟩, is a subgroup consisting of all powers of g.

Additional concepts include normal subgroups, which satisfy gHg^{-1} = H for all g ∈ G, and quotient groups G/H when H is normal. In finite groups, the order of a subgroup divides the order of the group (Lagrange’s theorem), and the index [G:H] counts cosets of H in G.