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 ). ${\ displaystyle \ exp (G)}$ ${\ displaystyle (G, \ cdot, e)}$ ${\ displaystyle n> 0}$ ${\ displaystyle g ^ {n} = e}$ ${\ displaystyle g}$ ${\ displaystyle G}$ ${\ displaystyle \ infty}$

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 . ${\ displaystyle (\ mathbb {Z} / n \ mathbb {Z}) ^ {\ times}}$

The group exponent of with a prime number is equal to the group order . ${\ displaystyle (\ mathbb {Z} / p \ mathbb {Z}) ^ {\ times}}$ ${\ displaystyle p}$ ${\ displaystyle p-1}$

The group exponent of is 2 (compare: The group order is 4).

{\ displaystyle (\ mathbb {Z} / 8 \ mathbb {Z}) ^ {\ times}}

The body with elements, understood as an additive group, has group order and group exponent (compare characteristics of a body ). ${\ displaystyle \ mathbb {F} _ {q}}$ ${\ displaystyle q = p ^ {k}}$ ${\ displaystyle q}$ ${\ displaystyle p}$

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. ${\ displaystyle \ mathbb {F} _ {p} [X]}$ ${\ displaystyle \ mathbb {F} _ {p}}$ ${\ displaystyle p}$

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 . ${\ displaystyle m / n + \ mathbb {Z}}$ ${\ displaystyle \ mathbb {Q} / \ mathbb {Z}}$ ${\ displaystyle n}$ ${\ displaystyle n> 0}$ ${\ displaystyle m}$ ${\ displaystyle n}$ ${\ displaystyle \ exp (\ mathbb {Q} / \ mathbb {Z}) = \ infty}$