group
Group.
A group is a pair $(G,*)$, where $G$ is a nonempty set and “$*$”
is a binary operation^{} on $G$, such that the following conditions hold:

•
For any $a,b$ in $G$, $a*b$ belongs to $G$. (The operation^{} “$*$” is closed).

•
For any $a,b,c\in G$, $(a*b)*c=a*(b*c)$. (Associativity of the operation).

•
There is an element $e\in G$ such that $g*e=e*g=g$ for any $g\in G$. (Existence of identity element^{}).

•
For any $g\in G$ there exists an element $h$ such that $g*h=h*g=e$. (Existence of inverses^{}).
If $G$ is a group under *, then * is referred to as the group operation of $G$.
Usually, the symbol “$*$” is omitted and we write $ab$ for $a*b$. Sometimes, the symbol “$+$” is used to represent the operation, especially when the group is abelian^{}.
It can be proved that there is only one identity element, and that for every element there is only one inverse. Because of this we usually denote the inverse of $a$ as ${a}^{1}$ or $a$ when we are using additive notation. The identity element is also called neutral element due to its behavior with respect to the operation, and thus ${a}^{1}$ is sometimes (although uncommonly) called the neutralizing element of $a$. An element of a group besides the identity element is sometimes called a nontrivial element.
Groups often arise as the symmetry groups of other mathematical objects; the study of such situations uses group actions^{}. In fact, much of the study of groups themselves is conducted using group actions.
Title  group 
Canonical name  Group 
Date of creation  20130322 11:42:53 
Last modified on  20130322 11:42:53 
Owner  drini (3) 
Last modified by  drini (3) 
Numerical id  34 
Author  drini (3) 
Entry type  Definition 
Classification  msc 14F99 
Classification  msc 08A99 
Classification  msc 20A05 
Classification  msc 2000 
Classification  msc 83C99 
Classification  msc 32C05 
Related topic  Subgroup^{} 
Related topic  CyclicGroup 
Related topic  Simple 
Related topic  SymmetricGroup 
Related topic  FreeGroup 
Related topic  Ring 
Related topic  Field 
Related topic  GroupHomomorphism 
Related topic  LagrangesTheorem 
Related topic  IdentityElement 
Related topic  ProperSubgroup 
Related topic  Groupoid^{} 
Related topic  FundamentalGroup 
Related topic  TopologicalGroup 
Related topic  LieGroup 
Related topic  ProofThatGInGImpliesThatLangleGRangleLeG 
Related topic  GeneralizedCyclicGroup 
Defines  identity^{} 
Defines  inverse 
Defines  neutralizing element 
Defines  nontrivial element 
Defines  nontrivial element 
Defines  group operation 