Difference between revisions of "Group (mathematics)"
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(Avoids the "positive-ness" issue, and still says what it should. Hooray for words!) |
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# the set of complex numbers {1, -1, <i>i</i>,<i>-i</i>} under multiplication, where <i>i</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 set of complex numbers {1, -1, <i>i</i>,<i>-i</i>} under multiplication, where <i>i</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 [[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. | ||
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 18:56, June 13, 2009
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.
Groups are the appropriate mathematical structures for any application involving symmetry.
