A group is a set paired with an operation on the set. As such, a group can be conceptualized as an ordered pair ${\displaystyle (G,\cdot)}$ , where ${\displaystyle G}$ is a set, and ${\displaystyle \cdot}$ is an operation.

A set and operation ${\displaystyle (G,\cdot)}$ is a group if and only if it satisfies the following properties:

1. Identity element — There exists an ${\displaystyle e\in G}$ , called an identity element, such that ${\displaystyle e\cdot g=g=g\cdot e}$ , for all ${\displaystyle g \in G}$
2. Inverses — For each ${\displaystyle g \in G}$ , there exists an ${\displaystyle h \in G}$ , called an inverse of ${\displaystyle g}$ , such that ${\displaystyle h\cdot g=e=g\cdot h}$
3. Associativity — For all ${\displaystyle a,b,c\in G\ ,\ (a\cdot b)\cdot c=a\cdot(b\cdot c)}$
4. Closure — For all ${\displaystyle a,b\in G\ ,\ a\cdot b\in G}$

Whenever the group operation is ${\displaystyle \cdot}$ , the operation of group elements ${\displaystyle a, b}$ , ${\displaystyle a\cdot b}$ , is often abbreviated as simply a juxtaposition of the group elements, ${\displaystyle ab}$ .

Important Results

From the given criterion for a group, the following properties can be shown for any group ${\displaystyle (G,\cdot)}$ :

• There exists exactly one identity element;
• For each ${\displaystyle g \in G}$ , there exists exactly one inverse of ${\displaystyle g}$ , and henceforth is referred to as ${\displaystyle g^{-1}}$ (proof)
• For each ${\displaystyle g\in G\ ,\ \left(g^{-1}\right)^{-1}=g}$
• Groups have the cancellation property: For all ${\displaystyle a,b,c\in G\ ,\ a\cdot b=a\cdot c}$ implies ${\displaystyle b=c}$ , and ${\displaystyle b \cdot a = c \cdot a}$ implies ${\displaystyle b=c}$ .

Optional Properties

A group ${\displaystyle (G,\cdot)}$ is:

• An abelian group if the operation ${\displaystyle \cdot}$ is commutative, i.e. ${\displaystyle ab = ba}$ for all ${\displaystyle a, b \in G}$
• A cyclic group if there exists a ${\displaystyle g \in G}$ such that ${\displaystyle G=\big\{g^n\big|n\in \Z\big\}}$ , where ${\displaystyle g^n}$ is ${\displaystyle n}$ copies of ${\displaystyle g}$ being operated together (see exponent)
• A subgroup of a group ${\displaystyle (K,\cdot)}$ if ${\displaystyle G\subseteq K}$ (see subset), where the group operation on ${\displaystyle G}$ is a domain restriction on the group operation on ${\displaystyle K}$

A group ${\displaystyle G}$ with a partial order ${\displaystyle \le}$ on it is a partially ordered group if for all ${\displaystyle a,b,g\in G}$ , if ${\displaystyle a \le b}$ , then ${\displaystyle ga\le gb}$ and ${\displaystyle ag\le bg}$ (translation invariance). It is a totally ordered group if in addition ${\displaystyle \le}$ is a total order.