Editing Exponentiation (section)

===In a group===
A [[multiplicative group]] is a set with as [[associative operation]] denoted as multiplication, that has an [[identity element]], and such that every element has an inverse.

So, if {{mvar|G}} is a group, <math>x^n</math> is defined for every <math>x\in G</math> and every integer {{mvar|n}}.

The set of all powers of an element of a group form a [[subgroup]]. A group (or subgroup) that consists of all powers of a specific element {{mvar|x}} is the [[cyclic group]] generated by {{mvar|x}}. If all the powers of {{mvar|x}} are distinct, the group is [[isomorphic]] to the [[additive group]] <math>\Z</math> of the integers. Otherwise, the cyclic group is [[finite group|finite]] (it has a finite number of elements), and its number of elements is the [[order (group theory)|order]] of {{mvar|x}}. If the order of {{mvar|x}} is {{mvar|n}}, then <math>x^n=x^0=1,</math> and the cyclic group generated by {{mvar|x}} consists of the {{mvar|n}} first powers of {{mvar|x}} (starting indifferently from the exponent {{math|0}} or {{math|1}}).

Order of elements play a fundamental role in [[group theory]]. For example, the order of an element in a finite group is always a divisor of the number of elements of the group (the ''order'' of the group). The possible orders of group elements are important in the study of the structure of a group (see [[Sylow theorems]]), and in the [[classification of finite simple groups]].

Superscript notation is also used for [[conjugacy class|conjugation]]; that is, {{math|1=''g''<sup>''h''</sup> = ''h''<sup>−1</sup>''gh''}}, where {{math|''g''}} and {{math|''h''}} are elements of a group. This notation cannot be confused with exponentiation, since the superscript is not an integer. The motivation of this notation is that conjugation obeys some of the laws of exponentiation, namely <math>(g^h)^k=g^{hk}</math> and <math>(gh)^k=g^kh^k.</math>