group
Group.
A group is a pair , where is a non-empty set and “”is a binary operation on , such that the following conditions hold:
- •
For any in , belongs to . (The operation
“” is closed).
- •
For any , . (Associativity ofthe operation).
- •
There is an element such that for any . (Existence of identity element
).
- •
For any there exists an element such that . (Existence of inverses
).
If is a group under *, then * is referred to as the groupoperation of .
Usually, the symbol “” is omitted and we write for. Sometimes, the symbol “” is used to represent theoperation, especially when the group is abelian.
It can be proved that there is only one identity element, and that forevery element there is only one inverse. Because of this we usuallydenote the inverse of as or when we are usingadditive notation. The identity element is also called neutralelement due to its behavior with respect to the operation, and thus is sometimes (although uncommonly) called the neutralizing element of . An element of a group besides theidentity element is sometimes called a non-trivial 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 | 2013-03-22 11:42:53 |
Last modified on | 2013-03-22 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 20-00 |
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 | non-trivial element |
Defines | nontrivial element |
Defines | group operation |