Periodic group

In the mathematical subfield of group theory is meant by the group Exponent of a group is the smallest natural number which applies to the ( power of a group element ). There is no such number, it is said, have exponent ( it must then also have infinite order ).

The existence of the (finite) group exponent of a group of finite order ensures Lagrange's Theorem.

To determine the exponent group of residue class groups the Carmichael function is used.

Properties

• By the theorem of Lagrange the group exponent is a divisor of the group order.
• In a cyclic group of exponent group with the group order is consistent.
• The group order is exactly right then match the group exponent when all Sylowgruppen the group are cyclic.
• The group exponent is the least common multiple (LCM ) of the order of all group elements.
• The group exponent of each subgroup is a divisor of the exponent of the group.

Examples

• The group of exponent for an arbitrary prime number is equal to the group order.
• The group of exponent 2 (see: The group order is 4).
• The body with elements taken as additive group has group order and group exponent (compare characteristics of a body).
• Infinite groups with finite exponent are, for example the polynomial and the algebraic closure of, respectively (because of prime characteristic ) in the additive link.
• Each element of the ( infinite), the torsion finite order n, where n > 0 and m and n are relatively prime. Since the element orders but are not limited, is.