undergrupper

A subgroup of a group G is a subset H that is itself a group under the same operation as G. A subset H of G contains the identity element of G, is closed under the group operation, and contains inverses for all its elements. If H is a subgroup of G, we write H ≤ G. If H is normal, we write H ⊴ G, enabling the construction of the quotient group G/H.

Examples include the additive group of integers (Z, +) with the even integers 2Z forming a subgroup; the multiplicative group of nonzero real numbers (R*, ×) or the positive real numbers R^+ forming subgroups of (R, ×); and the trivial subgroup {0} in additive groups. Subgroups can be finite or infinite, and any set of generators S yields the smallest subgroup containing S, denoted ⟨S⟩.

Key properties and related concepts: for a finite group G, the order of any subgroup H divides the order of G (Lagrange’s theorem). The cosets gH partition G, and normal subgroups give rise to quotient groups G/H, which reflect the structure of G modulo H. The kernel of a homomorphism φ: G → K is a subgroup of G, and φ induces an isomorphism between G/kerφ and imφ (the First Isomorphism Theorem). Subgroups are central to many areas of algebra and appear in topics such as group actions, symmetry, and Galois theory.