Group

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

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

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

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

Important Results

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

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

Optional Properties

A group $(G,\cdot)$ is:

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

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