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Group

Definition

A group is a set $G$ equipped with a binary operation $*$, that obeys the following axioms:

• The operation is associative, i.e. $\forall g,h,i \in G: (g*h)*i=g*(h*i)$
• There is an identity element $i$ s.t. $\forall g \in G: i*g = g*i$.
• Each element has an inverse, i.e. $\forall g \in G: \exists g^{-1} \in G : g*g^{-1} = i = g^{-1}*g$. This g^{-1} is called the inverse of g.

Properties

The order of the group is the number of elements in the group.

Types

If the operation is commutative we say G is an "Abelian" or "commutative" group

If the group is generated by a single element g in G, we say G is "cyclic".

Examples

The units of a ring form a group.