Notation $\pi(G)$ for a group $G$

Question

For a finite group $G$, I've encountered the notation $\pi(G)$ that is the set of all primes that divide the order of $G$.

If the group $G$ is infinite, what does $\pi(G)$ mean?

Answer 1

I am not a group theorist, so I don't know if there is a standard meaning for this notation.

With that said, a nice fact about finite groups is that a prime $p$ divides the order of $G$ if and only if $G$ has an element of order $p$. So you could use $\pi(G)$ to represent the set of primes $p$ such that $G$ has an element of order $p$. This makes sense for all groups, and agrees with the definition you gave for finite groups.

You may also want to have $0 \in \pi(G)$ if $G$ has elements of infinite order.

Answer 2

If $\vert G\vert = \infty $ there are no primes that divide the order of $G$ so $\pi (G) = \emptyset $.