twogenerator

Two-generator is a term used in algebra to describe a group that can be generated by two elements. More precisely, a group G is two-generator if there exist elements a and b in G such that every element of G can be expressed as a product of powers of a and b. This notion is often stated as the group having a generating set of size at most two, which includes cyclic groups as a special case.

Examples of two-generator groups include the symmetric group S3, which can be generated by a transposition and a 3-cycle, such as (12) and (123). The dihedral group D4, the symmetry group of a square, is generated by a rotation r and a reflection s. In contrast, a cyclic group generated by a single element is also two-generator in the sense that its generating set can be reduced to a two-element set if needed. Finite abelian groups and many other finite groups are two-generator, though some require more than two elements to generate.

The concept is closely linked to presentations and free groups. A two-generator group can be described by a presentation of the form G ≅ ⟨a, b | R⟩, where R is a set of relations among a and b. The free group on two generators, F2, is the most general source of two-generator quotients: every two-generator group is a quotient of F2 by a normal subgroup determined by the relations. This perspective is central in combinatorial and computational group theory.

In topology and geometry, two generators often suffice to capture the essential structure of a group, as seen in fundamental groups of surfaces (such as the torus) and in various symmetry groups. The study of two-generator groups intersects with questions about minimal generating sets, group presentations, and algorithms for constructing and recognizing groups.