Exponent of subgroup divides exponent of group

Definition

Suppose $G$ is a group and $H$ is a subgroup of $G$. Then, the exponent of $H$ divides the exponent of $G$.

In particular, this means that if the exponent of $G$ is finite, then so is the exponent of $H$. Conversely, if the exponent of $H$ is infinite, so is the exponent of $G$.

Related facts

• Lagrange's theorem
• Exponent of quotient group divides exponent of group
• Exponent of extension group is a multiple of lcm of exponents of normal subgroup and quotient group
• Exponent of extension group divides product of exponents of normal subgroup and quotient group
• Exponent of direct product is lcm of exponents

Proof

This is by definition: the exponent of the subgroup is the lcm of the orders of all elements in the subgroup. The exponent of the whole group is the lcm of the orders of all elements in the group, which is a possibly larger set of numbers. Thus, the former divides the latter.