Exponent of a finite group has precisely the same prime factors as order

Statement

For a finite group, the set of prime numbers that divide the order of the group is the same as the set of prime numbers that divide the exponent of the group.

Related facts

• Exponent divides order
• Exponent of a finite group equals product of exponents of its Sylow subgroups

Facts used

1. Exponent divides order
2. Cauchy's theorem: If $p$ divides the order of a finite group $G$, there is an element of $G$ with order exactly $p$.

Proof

By fact (1), the exponent of a finite group divides its order, so any prime factor of the exponent is a prime factor of the order. By fact (2), the exponent must be a multiple of each prime factor of the order of the group. Combining these, we obtain that the set of prime factors of the exponent is the same as the set of prime factors of the order.