FANDOM


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:

  • 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.