Group

A group is a monoid $(G, \cdot, e)$ in which every element $g$ has an inverse $g^{-1}$ satisfying $g \cdot g^{-1} = e = g^{-1} \cdot g$. Equivalently, a group is a set equipped with an associative binary operation, an identity element, and inverses for every element.

The inverse of each element is unique, as is the identity. These facts follow from the full group axioms (associativity, existence of identity, and existence of inverses together) and do not need to be assumed separately.

In category theory, a group is a category with a single object in which every morphism is an isomorphism. This perspective connects groups directly to the vault’s categorical framework.

Groups are the simplest algebraic structures with a notion of reversibility. They form the foundation of abstract algebra and appear throughout mathematics as the language of symmetry.