Groupswithout

Groupswithout is a term encountered in discussions of group theory to describe a way of focusing on subgroups that can be formed without using all elements of a given generating set. While not part of standard textbooks, the concept appears in problem collections and informal expositions as a means to study how much of a group’s structure is captured by partial generating information.

Definition. Let G be a group with a fixed generating set X. The groupswithout construction, denoted groupswithout(G,X), is the collection of subgroups H ≤ G for which there exists a nonempty proper subset Y ⊆ X such that H = ⟨Y⟩. In other words, it is the family of subgroups that can be generated by omitting at least one generator from X. The elements are ordered by inclusion, forming a poset.

Examples. If G = ⟨a,b⟩ is the free group on two generators and X = {a,b}, then groupswithout(G,X) contains ⟨a⟩ and ⟨b⟩, since each is generated by a single generator and uses a proper subset of X. The full group ⟨a,b⟩ is typically excluded by the nonempty proper-subset requirement, though conventions may vary and some treatments include the trivial subgroup ⟨⟩ generated by the empty subset.

Properties and use. The construction highlights which subgroups arise from subsets of generators and helps illustrate the dependence of subgroups on generating information. It is useful in teaching to visualize how the subgroup lattice relates to the generating set and in computational group theory as a way to organize subset-generated candidates for subgroups.

See also. Subgroups, Generating sets, Lattice of subgroups, Quotient groups.