I tried to answer the following exercise:
Reformulate the corollary of Theorem 4.4. to include the case when the group has infinite order.
The corollary in question is this:
In a finite group the number of elements of order $n$ is divisble by $\varphi(n)$ where $\varphi$ is the totient function.
In an infinite group the number of elements of order $n$ is alse divisible by $\varphi(n)$.
Is this correct?