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 elementThere exists an e\in G , called an identity element, such that e\cdot g=g=g\cdot e , for all g\in G
  2. InversesFor 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:

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.