Subgroup

A subgroup of a group G is a subset H ⊆ G that is itself a group under the same operation as G. Equivalently, H ≤ G means H is nonempty and closed under the group operation and taking inverses; in particular, for any a,b ∈ H we have ab ∈ H and a^{-1} ∈ H, which also implies the identity e ∈ H.

Subgroups can be identified by the subgroup test: a nonempty subset H of G is a subgroup if it is closed under the group operation and under taking inverses. The order of a subgroup of a finite group divides the order of the group (Lagrange’s theorem).

Examples illustrate the concept. The set of even integers 2Z is a subgroup of the integers Z under addition. The alternating group A_n is a subgroup of the symmetric group S_n.

Subgroups have structural properties. The intersection of any collection of subgroups of G is again a subgroup. For any subset S ⊆ G, the subgroup generated by S, denoted ⟨S⟩, is the smallest subgroup of G containing S. A subgroup that is not equal to G is a proper subgroup. The trivial subgroup {e} contains only the identity element.

Normal subgroups and quotients are central in group structure. A subgroup N ≤ G is normal if gNg^{-1} = N for all g ∈ G, and such subgroups yield the quotient group G/N. In general, the union of subgroups need not be a subgroup, and the product of two subgroups is a subgroup only in special cases (e.g., when one is contained in the other or when one is normal).