Group

Definition

Category-theoretic definition

A group is a small category with one object where all morphisms are isomorphisms. (note that the small assumption is to ensure that the morphisms form a set; however, it can be dropped for some purposes).

Ordinary definition

A group is a set $G$ equipped with a binary operation $*$, a unary operation ${}^{-1}$, and a constant $e$ such that the following hold:

• Associativity: $a*(b*c)=(a*b)*c\ \forall \ a,b,c\in G$.
• Neutral element or identity element: $a*e=e*a=a\ \forall \ a\in G$.
• Inverse element: $a*a^{-1}=a^{-1}*a=e$.

(there are other slightly different formulations of this definition).

Definition building on monoid

A group is a monoid in which every element has a two-sided inverse.

Equivalence of definitions

The two definitions are equivalent in the following sense: given any group in the category-theoretic set, the morphisms, under composition, form a group in the set-theoretic sense. Conversely, given any group in the set-theoretic sense, we can construct a category with one object and with the morphisms corresponding to the elements of the group.

Further, this equivalence gives an equivalence of categories, where the first category is the subcategory corresponding to groups of the category of locally small categories (so the morphisms are functors), and the second category is the category of groups in the conventional sense.

External links

Other subject wikis

• Group on the Group Properties Wiki -- the most detailed and canonical reference.
• Group on the Topology Wiki

Other resources

• group on nlab
• Group on Wikipedia