Where does group collide?

A set equipped with a single binary operation that combines any two of its elements into a third, subject to four axioms: the operation is closed on the set, it is associative, there is an identity element that leaves every element unchanged, and every element has an inverse that returns the identity. Dummit and Foote's Abstract Algebra gives this as the founding object of group theory, the abstract structure that captures the symmetries of an object under composition. The referent is a formal algebraic structure, a set together with a law, not a body of people, a mental category, or a cluster of atoms in a molecule.[1]

Unit

A set closed under one associative operation with identity and inverses

Recovery means

Not applicable

Measured by

Established by checking the four axioms, and characterised by its order, its subgroups, and the way it acts as symmetries

A number of people who interact with one another over time, who are aware of belonging together, and who share some sense of unity or common identity. Homans's The Human Group treated the group as a system of recurring activities, interactions, and sentiments among its members, the elementary unit of social life. The referent is a concrete set of people bound by real interaction and mutual recognition, and it is set apart from a mere aggregate of co-present strangers or a statistical category that never meets, and apart from an operation on a set or a piece of a molecule.[2]

Unit

A set of people bound by ongoing interaction and shared identity

Recovery means

Not applicable

Measured by

Studied through observation of interaction, membership rolls, sociometric ties, and members' expressed sense of belonging

The cognitive division of the social world into an ingroup that a person counts as "us" and an outgroup counted as "them," a categorisation that shapes perception, evaluation, and behaviour even before any real relationship exists. Tajfel, Billig, Bundy, and Flament's minimal group experiments showed that assigning people to bare, meaningless categories was enough to make them favour their own side in allocating rewards. The referent is a psychological act of self and other categorisation carried in the mind, not a standing group of interacting people, an algebraic structure, or a reactive cluster of atoms.[3]

Unit

A mental categorisation of self and others into us and them

Recovery means

Not applicable

Measured by

Probed with the minimal group paradigm, allocation matrices, and measures of ingroup bias and identification

An atom or a defined arrangement of atoms within a molecule that carries similar chemical properties wherever it occurs, so that it governs the characteristic reactions of the compounds that contain it. The IUPAC definition treats the functional group, such as a hydroxyl or a carbonyl, as the reactive part that sits on a relatively inert backbone and defines a family of organic compounds. The referent is a substructure of bonded atoms responsible for reactivity, not a set with an operation, a body of people, or a mental us and them.[4]

Unit

A substructure of atoms conferring characteristic reactivity on a molecule

Recovery means

Not applicable

Measured by

Identified by spectroscopy such as infrared and NMR, and by the characteristic reactions the group undergoes