Order (group theory)

In the mathematical subfield of group theory is meant by the order of a group element or element order of an element of a group is the smallest natural number for which holds, where 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. The order is sometimes referred to, with or.

For this, we define the power of a group element:

• With

The number, if it is finite, called group exponent.

Properties

• By the theorem of Lagrange all the elements of a finite group has a finite order, ie a divisor of the group order, the number of elements of the group.
• Conversely exists after the Sylow - sets for every prime divisor of the group order in a finite group an element that has the atomic. For dividers, that are not prime, no general statement is possible.
• The order of an element is equal to the order of the group that is generated by this element.
• It can be shown that the following applies.
• In Abelian groups, the order of the product is a divisor of the least common multiple of the orders of and. In non- Abelian groups, no such statement is possible; For example, the element of the group SL2 (Z ) has infinite order, but it is equal to the product of the elements and the respective orders 4 and 6

• Group Theory