subgroups

In abstract algebra, a subgroup H of a group G is a subset that is itself a group under the same operation as G. Equivalently, H is a subgroup if it contains the identity element of G, is closed under the group operation, and contains inverses of its elements. Because the identity, products, and inverses in H are inherited from G, many tests reduce to checking these closure properties for elements of H.

Examples: The trivial subgroup {e} and the whole group G are subgroups of G. For any g in G, the cyclic subgroup generated by g, written <g>, consists of all integer powers of g and is a subgroup of G. In the additive setting, for instance the subgroups nZ of the integers form subgroups of (Z,+). Subgroups can be finite or infinite, and may be proper or equal to G.

Notable properties: The intersection of any family of subgroups is a subgroup. Given a subset S of G, the intersection of all subgroups containing S equals the subgroup generated by S. A subgroup H is normal if gHg^{-1} = H for all g in G; normal subgroups yield quotient groups G/H. The kernel of a group homomorphism is a normal subgroup, while the image is a subgroup of the codomain.

In finite groups, Lagrange's theorem asserts that the order of a subgroup divides the order of the group; the index [G:H] equals the number of left cosets. Subgroups play a central role in structure theorems and constructions, such as direct products, semidirect products, and extensions.