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.
My answer:
In an infinite group the number of elements of order $n$ is alse divisible by $\varphi(n)$.
Is this correct?
Your answer doesn't make sense if the number of elements of order $n$ is infinite (or rather, your answer is vacuous in that case).
My guess is the following reformulation is the intended answer: If $G$ is a group, and $T_n$ is the set of elements of order $n$ in $G$, then the sets $\{g^k \mid \gcd(k, n)=1\}$ for $g \in T_n$ form a partition of $T_n$ into subsets of size $\varphi(n)$.