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Group

Definition

A group is a set $G$ equipped with an associative binary operation $*$, such that:

• 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.

The operation is also called the group law.

The identity is also called the neutral element.

We usually use $G$ to denote the group, rather than $(G,*)$.

The operation is usually denoted with a dot, like multiplication. The identity element is

Properties

The inverse of any element is unique.

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 $R^\times$ of a ring $R$ form a group.

Given groups $G,H$, the direct sum $G \oplus H$ is the set of ordered pairs $(g,h): g \in G, h \in H$ with the operation being a combination of the operations from $G$ and $H$.

all these pages adapted with probably insufficient credit from my university's lecture notes