Group exponent
In the mathematical sub-area of group theory , the group exponent of a group is understood to be the smallest natural number for which ( power of a group element ) applies to all group elements . If there is no such number, it is said that it has an exponent (it must then also be of infinite order ).
properties
- According to Lagrange's theorem , the group exponent for a finite group is a divisor of the group order and therefore in particular finite.
- In a cyclic group of torsion group with the right group order the same.
- The group order exactly matches the group exponent if all Sylow groups in the group are cyclic.
- The group exponent is the smallest common multiple (LCM) of the order of all group elements.
- The group exponent of a subgroup is a factor of the exponent of the whole group.
Examples
- For the prime residual class groups , the group exponent is obtained from the Carmichael function .
- The group exponent of with a prime number is equal to the group order .
- The group exponent of is 2 (compare: The group order is 4).
- The body with elements, understood as an additive group, has group order and group exponent (compare characteristics of a body ).
- Infinite groups with finite exponents are, for example, the polynomial ring and the algebraic closure of , each (because of the prime number characteristic ) in the additive combination.
- Every element of the (infinite) torsion group has finite order if it holds and is too coprime . Since the element orders are not restricted, is .
See also
Individual evidence
- ^ Wikiversity. Retrieved August 13, 2012.
- ↑ Matroids Math Planet. Post No. 7 from Gockel. Retrieved August 13, 2012.