nearcommutativity

Nearcommutativity is a concept in abstract algebra that describes elements in a structure, such as a group or a ring, that "almost commute" with each other. This means that for two elements a and b, the product ab is not equal to ba, but the difference or some related measure is "small" or belongs to a specific subset of the structure.

In the context of a group G, two elements a and b are said to be near-commuting if their commutator, defined as [a, b] = ab a⁻¹b⁻¹, is equal to the identity element or belongs to a proper subgroup H of G. If [a, b] is always in H for all a, b in G, then G is said to be H-near-commuting. For instance, if H is the center of the group, then H-near-commuting is equivalent to commutativity.

The concept has found applications in various areas of mathematics, including the study of solvable groups and the classification of certain algebraic structures. In some contexts, nearcommutativity can be a weaker condition than full commutativity, allowing for a richer class of structures to be studied. It provides a way to quantify how far an algebraic system deviates from being commutative. Research in nearcommutativity often involves understanding the properties of elements that satisfy this condition and the implications for the overall structure.