# Changes

## 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 $A$, there must exist an inverse $A^{-1}$ such that $AA^{-1} = A^{-1}A = I$
A group with [[commutative ]] binary operator is known as [[Abelian group|Abelian]].
Example 1==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 $Z_\mathbb{Z}_{2}\times Z_\mathbb{Z}_{2}$ 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 $Z_{4}$ under mod addition. Groups are the appropriate mathematical structures for any application involving [[symmetry]].
[[Category:Algebra]]
[[category:mathematics]]