subgroupen

Subgroupen is the Dutch-language term used to refer to subgroups in group theory. It denotes subsets of a group that themselves form a group under the same operation as the ambient group.

Formal definition: Let (G, *) be a group. A subset H of G is a subgroup if it contains the identity element e of G, is closed under the operation (for any a,b in H, a * b is in H), and contains inverses (for any a in H, a^{-1} is in H). Equivalently, H is a subgroup if it is nonempty and closed under the product and taking inverses, or if for all a,b in H, a * b^{-1} is in H.

Subgroup tests and examples: A common test is the subgroup criterion, which states that for all a,b in H, a * b^{-1} is in H. Examples include the set nZ of multiples of an integer n within the group (Z, +), the trivial subgroup {e}, and the whole group G itself. In the symmetric group S_n, the alternating group A_n is a subgroup.

Lattice structure and generation: Subgroups of a given group form a lattice under inclusion. The intersection of any collection of subgroups is a subgroup, and for any subset S of G, the subgroup generated by S, denoted ⟨S⟩, is the smallest subgroup containing S.

Normal subgroups and quotients: A subgroup N is normal if gNg^{-1} = N for all g in G, enabling construction of the quotient group G/N. Normal subgroups play a central role in many theorems, including those about homomorphisms and factor groups.

Notes on terminology: In Dutch mathematics, the standard term is subgroep with the plural subgroepen. The form subgroupen may appear in some texts as an Anglicized or informal variant. See also subgroep, subgroup, and group theory.