section_path: Integer powers in algebra/In a group
section_name: In a group
paragraph_text: 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 G is a group, x n {\displaystyle x^{n}} is defined for every x ∈ G {\displaystyle x\in G} and every integer 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 x is the cyclic group generated by x. If all the powers of x are distinct, the group is isomorphic to the additive group Z {\displaystyle \mathbb {Z} } of the integers. Otherwise, the cyclic group is finite (it has a finite number of elements), and its number of elements is the order of x. If the order of x is n, then x n = x 0 = 1, {\displaystyle x^{n}=x^{0}=1,} and the cyclic group generated by x consists of the n first powers of x (starting indifferently from the exponent 0 or 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 conjugation; that is, g = h gh, where g and 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 (g h) k = g h k {\displaystyle (g^{h})^{k}=g^{hk}} and (g h) k = g k h k. {\displaystyle (gh)^{k}=g^{k}h^{k}.}