# 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}$

## 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}$ ${\ displaystyle \ exp (\ mathbb {Q} / \ mathbb {Z}) = \ infty}$