Exponent divides order in finite group
From Groupprops
(Redirected from Exponent divides order)
Statement
In a finite group, the exponent, which is defined as the least common multiple of the orders of all the elements of the group, divides the order of the group.
Related facts
Converse
- Cauchy's theorem: This states that there is an element of prime order
for every prime
dividing the order of the group.
- Exponent of a finite group has precisely the same prime factors as order
Facts used
Proof
The proof follows directly from fact (1) and the definition of exponent as the least common multiple of the orders of individual elements.