I am wondering if there is any infinite non abelian group with exactly $ \phi (n) $ elements of order $n$ for every $ n $
If not, is there any thing like if for each $ n $ the group contains $ \phi (n) $ elements of order $n$ then the group is abelian?
Just some random thoughts. Does anyone have any examples? If not, then do you have any tool to prove the later part? Please don't disclose the proof just let me have some hints.
Thanks.