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Group (mathematics)

952 bytes added, 13:41, 13 July 2016
/* Examples */clean up & uniformity
{{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
#'''Existence of Inverse''': for each element <math>A</math>, there must exist an inverse <math>A^{-1}</math> such that <math>AA^{-1} = A^{-1}A = I</math>
A group with [[commutative ]] binary operator is known as [[Abelian group|Abelian]].
Example 1==Examples==# 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 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>. # 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 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> Z_\mathbb{Z}_{2}\times Z_\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.
Example 2: the set of complex numbers {1, -1, <i>i</i>,<i>-i</i>} under multiplication, where <i>i</i> is the square root of -1, the basis of the [[imaginary number]]s. This group is [[isomorphism|isomorphic]] to <math> Z_{4} </math> under mod addition. Groups are the appropriate mathematical structures for any application involving [[symmetry]].
[[Category:Algebra]]
[[category:mathematics]]
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