# Order of a group element

In the mathematical sub-area of group theory , the **order of a group element** or **element order of** an element of a group is understood to be the smallest natural number for which applies, where is the neutral element of the group. There is no such number, it is said, have *infinite order. *Elements of finite order are also called torsion elements. The order is sometimes referred to as or .

The power of a group element is inductively defined for natural exponents :

- for all natural

If the number is finite, it is called the group exponent .

## properties

- According to Lagrange's theorem , all elements of a finite group have a finite order, and this is a factor of the group order , i. H. the number of elements in the group.
- Conversely, in a finite group, according to Cauchy's theorem, for every prime divisor of the group order there is an element that has the order . No general statement is possible for compound factors (while the neutral element belongs to the trivial factor 1 ).
- The order of an element is equal to the order of the subgroup , of this element generates is.
- It applies if and only if is a multiple of the order of the element .
- In Abelian groups , the order of the product is a factor of the least common multiple of the orders of and . No such statement is possible in non-Abelian groups; for example, the element of group SL
_{2}(**Z**) has infinite order, although it is the product of the elements of order 4 and of order 6._{}

## literature

- JC Jantzen, J. Schwermer:
*Algebra.*Springer, Berlin / Heidelberg 2006, ISBN 3-540-21380-5 .