Exponent divides order in finite group

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 $p$ for every prime $p$ dividing the order of the group.
• Exponent of a finite group has precisely the same prime factors as order

Facts used

1. Order of element divides order of group

Proof

The proof follows directly from fact (1) and the definition of exponent as the least common multiple of the orders of individual elements.