stylistic:style_transfer, stylistic:tone
Groups with additional structure

An equivalent definition of group consists of replacing the "there exist" part of the group axioms by operations whose result is the element that must exist. So, a group is a set G {\displaystyle G} equipped with a binary operation G × G → G {\displaystyle G\times G\rightarrow G} (the group operation), a unary operation G → G {\displaystyle G\rightarrow G} (which provides the inverse) and a nullary operation, which has no operand and results in the identity element. Otherwise, the group axioms are exactly the same. This variant of the definition avoids existential quantifiers and is used in computing with groups and for computer-aided proofs.

This way of defining groups lends itself to generalizations such as the notion of group object in a category. Briefly, this is an object with morphisms that mimic the group axioms.