Difference between revisions of "Group (mathematics)"
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| − | A '''group''' is a mathematical structure consisting of a set of elements combined with a binary operator which satisfies four conditions: | + | {{Math-h}} |
| + | A '''group''' is a mathematical structure consisting of a [[set]] of elements combined with a [[binary operator]] which satisfies four conditions: | ||
#'''Closure''': applying the binary operator to any two elements of the group produces a result which itself belongs to the group | #'''Closure''': applying the binary operator to any two elements of the group produces a result which itself belongs to the group | ||
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==Examples== | ==Examples== | ||
| − | # the set of [[integers]] <math>\mathbb{Z}</math> under addition: here, zero is the identity, and the inverse of an element <math>a \in \mathbb{Z}</math> is <math>-a</math>. | + | # the set of [[integers]] <math>\mathbb{Z}</math> under addition, <math>(\mathbb{Z},+)</math>: here, zero is the identity, and the inverse of an element <math>a \in \mathbb{Z}</math> is <math>-a</math>. |
| − | # the set of the positive [[rational | + | # the set of the positive [[rational number]]s <math>\mathbb{Q}_+</math> under multiplication, <math>(\mathbb{Q}_+,\cdot)</math>: <math>1</math> is the identity, while the inverse of an element <math>\frac{m}{n} \in \mathbb{Q}_+</math> is <math>\frac{n}{m}</math>. |
| − | # the Klein four group consists of the set of formal symbols <math>\{1, i, j, k \} </math> with the relations <math> i^{2} =j^{2}=k^{2}=1, \; ij=k, \; jk=i, \; ki=j. </math> All elements of the Klein four group (except the identity 1) have [[order]] 2. The Klein four group is [[isomorphism|isomorphic]] to <math>\mathbb{Z}_{2} \times \mathbb{Z}_{2}</math> under mod addition. | + | # for every <math>n \in \mathbb{N}</math> there exists at least one group with n elements,e.g., <math>(\mathbb{Z}/n\mathbb{Z},+) = (\mathbb{Z}_n,+). </math> |
| − | # the set of | + | # the set of complex numbers {1, -1, ''i'',''-i''} under multiplication, where ''i'' is the principal square root of -1, the basis of the [[imaginary number]]s. This group is [[isomorphism|isomorphic]] to <math> \mathbb{Z}_{4} </math> under mod addition. |
| + | # the [[Klein four group]] consists of the set of formal symbols <math>\{1, i, j, k \} </math> with the relations <math> i^{2} =j^{2}=k^{2}=1, \; ij=k, \; jk=i, \; ki=j. </math> All elements of the Klein four group (except the identity 1) have [[order]] 2. The Klein four group is [[isomorphism|isomorphic]] to <math>\mathbb{Z}_{2} \times \mathbb{Z}_{2}</math> under mod addition. | ||
| + | # the set of "moves" on a Rubik's cube, where a move is understood to be a finite sequence of twists: here, the identity move is to do nothing, while the inverse of a move is to do the move in reverse, thereby undoing it. | ||
| + | # The [[Symmetric group]] | ||
| + | # The general and special [[Linear group]]s. | ||
Groups are the appropriate mathematical structures for any application involving [[symmetry]]. | Groups are the appropriate mathematical structures for any application involving [[symmetry]]. | ||
[[Category:Algebra]] | [[Category:Algebra]] | ||
Revision as of 13:41, July 13, 2016
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This article/section deals with mathematical concepts appropriate for late high school or early college. |
A group is a mathematical structure consisting of a set of elements combined with a binary operator which satisfies four conditions:
- Closure: applying the binary operator to any two elements of the group produces a result which itself belongs to the group
- Associativity:
where 
, 
and 
are any element of the group - Existence of Identity: there must exist an identity element
such that 
; that is, applying the binary operator to some element and the identity element leaves unchanged - Existence of Inverse: for each element , there must exist an inverse
such that 
A group with commutative binary operator is known as Abelian.
Examples
- the set of integers
under addition, 
: here, zero is the identity, and the inverse of an element 
is 
. - the set of the positive rational numbers
under multiplication, 
: 
is the identity, while the inverse of an element 
is 
. - for every
there exists at least one group with n elements,e.g., 
- the set of complex numbers {1, -1, i,-i} under multiplication, where i is the principal square root of -1, the basis of the imaginary numbers. This group is isomorphic to
under mod addition. - the Klein four group consists of the set of formal symbols
with the relations 
All elements of the Klein four group (except the identity 1) have order 2. The Klein four group is isomorphic to 
under mod addition. - the set of "moves" on a Rubik's cube, where a move is understood to be a finite sequence of twists: here, the identity move is to do nothing, while the inverse of a move is to do the move in reverse, thereby undoing it.
- The Symmetric group
- The general and special Linear groups.
Groups are the appropriate mathematical structures for any application involving symmetry.
