# Changes

## Group (mathematics)

527 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:
==Examples==
# the set of [[integers]] $\mathbb{Z}$ under addition, $(\mathbb{Z},+)$: here, zero is the identity, and the inverse of an element $a \in \mathbb{Z}$ is $-a$. # the set of the positive [[rational number]]s $\mathbb{Q}_+$ under multiplication, $(\mathbb{Q}_+,\cdot)$: $1$ is the identity, while the inverse of an element $\frac{m}{n} \in \mathbb{Q}_+$ is $\frac{n}{m}$. # for every $n \in \mathbb{N}$ there exists at least one group with n elements,e.g., $(\mathbb{Z}/n\mathbb{Z},+) = (\mathbb{Z}_n,+).$# 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 $\mathbb{Z}_{4}$ under mod addition.
# the [[Klein four group]] consists of the set of formal symbols $\{1, i, j, k \}$ with the relations $i^{2} =j^{2}=k^{2}=1, \; ij=k, \; jk=i, \; ki=j.$ All elements of the Klein four group (except the identity 1) have [[order]] 2. The Klein four group is [[isomorphism|isomorphic]] to $\mathbb{Z}_{2} \times \mathbb{Z}_{2}$ 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 -1twists: 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 numberSymmetric group]]s. This group is # The general and special [[isomorphism|isomorphicLinear group]] to $\mathbb{Z}_{4}$ under mod additions.
Groups are the appropriate mathematical structures for any application involving [[symmetry]].
[[Category:Algebra]]