# Exponent divides order in finite group

From Groupprops

## 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.