# Changes

/* 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:

==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>\mathbb{Z}_{2} \times \mathbb{Z}_{2}</math> under mod addition.

# the set of ~~complex numbers {1, -1, <i>i</i>,<i>-i</i>} under multiplication~~"moves" on a Rubik's cube, where ~~<i>i</i> ~~a move is ~~the square root ~~understood to be a finite sequence of ~~-1~~twists: here, the ~~basis ~~identity move is to do nothing, while the inverse of a move is to do the move in reverse, thereby undoing it.# The [[~~imaginary number~~Symmetric group]]~~s. This group is ~~# The general and special [[~~isomorphism|isomorphic~~Linear group]] ~~to <math> \mathbb{Z}_{4} </math> under mod addition~~s.

Groups are the appropriate mathematical structures for any application involving [[symmetry]].

[[Category:Algebra]]