This is a theorem of Rosenstein from the paper *[$\aleph_0$-categoricity of groups][1]*. 
Often the terminology "finite order" means "finite cardinality" in the setting of groups, so to avoid confusion I will borrow Rosenstein's terminology instead: A group $G$ has *bounded order* if there is some integer $n$ such that $g^n=1$ for all $g\in G$.  

I also say that a group is *$\aleph_0$-categorical* if its complete theory in the language of groups is $\aleph_0$-categorical (i.e., has a unique countable model). 

It is not hard to show that an $\aleph_0$-categorical group has bounded order. This is also Theorem 1 of Rosenstein's paper. Theorem 2 of his paper provides a converse for abelian groups. So we have:

>**Theorem.** (Rosenstein 1971)<br/>
>An infinite abelian group  is $\aleph_0$-categorical if and only if it has bounded order.


As a side note, I had not heard the terminology "totient group" used this way before. 




  [1]: https://pdf.sciencedirectassets.com/272332/1-s2.0-S0021869300X03328/1-s2.0-0021869373900926/main.pdf?X-Amz-Security-Token=IQoJb3JpZ2luX2VjEO7%2F%2F%2F%2F%2F%2F%2F%2F%2F%2FwEaCXVzLWVhc3QtMSJHMEUCIHZ04EYzAqky510u1oBygW1At1j7x%2B5r3OV5TdolDRWPAiEA5ivtoZxXofL7aNGuQV2i3JLgT5h2rBKfjeb3thSqnpMqvQMIlv%2F%2F%2F%2F%2F%2F%2F%2F%2F%2FARADGgwwNTkwMDM1NDY4NjUiDL%2BDlH2eCwH7WlCJGSqRA%2BswnBQPE5BKYbaRV2PyQ4aGGv5yanbZr%2BNsSeoh7bzkFnaCOBldZUi3QTAbvATcNeQ%2B5U3nhtK%2Bk1T9XAPK01KK%2BJz2CI4%2BbO9MexuSIFM51tIpIw4HGAlqKh0Vivde20HjMftCSsu2SnHVpAm%2F9Qmq4IA%2Bl0y9SCUpRAjEnHIv6KMXDK%2Ba8EtBTjKgEy%2FDr3BA3Yl9V6LgdQ%2B1nCLnIHTwU0OEzLH2NHg9U4%2FzAm1H%2Bijj0P1ZzH2iE%2FnhNrtLDFijudB0TBDdRvlHPuS%2BzXcG3GYwxpvXkAQOCHjSHrcXOSpq5U6H%2BIdGU5KRSHT3NQrWg8SrqmfSTHfgexwzaTzjIug7aCjPxcfwjM4pISJs5l0gIYxO7mvYRspQSAQ6AEUheMjA%2B7X4lA4LZkDCUqFiAeAj3XuwGBcFSW0gl%2FITLqj%2FlBLLm1pFYnCp5%2BJBbCwBtjNoECW%2Br1nT4siVY5Hw356%2ByI11jZ1ki3nddALe%2FJKftuQfZda%2F%2BbAww1vUFfgsXO3Act4WTwuq7w%2FyTEnLMInBuPgFOusBxl1bgaKdr6n6asqTO6i7L4enprFVlt50OxnckSPq8FXKPtuF%2FPCv6in%2BEFAruPqy4L6otfG7XRusxiYl6Nal6yWgomSVkqkj8CNf67%2BfIC4yA2voGnyWT1RbS5Ii8zerGCWNVKJFKQTdSUscpT%2B0VN06tAPKHux7QrfSx%2Fav%2BNiIfWJ%2Bx1hOEg7lNGgXOqwB2zZ0dHdddrgdpGtSJ%2Bj6qO%2F9vC%2BAbqFsNLUk9jfa5DQefkhswtRZqEuWn9E4C2qalYDkpJvSqFtOE0jfWA8ITQXZoLaGmAc%2F7v4FKEhKkFHOma0mYkEmfX92rQ%3D%3D&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20200714T223940Z&X-Amz-SignedHeaders=host&X-Amz-Expires=300&X-Amz-Credential=ASIAQ3PHCVTYTHJVZKX3%2F20200714%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Signature=a7ed2b321ded4639a7ae34495eb16ebd013b44f97aabdadd8add96cb93031b9b&hash=379d723ce72c0430fb0fccaa6082460b05aa68b6845798676ca4b968e3ec62aa&host=68042c943591013ac2b2430a89b270f6af2c76d8dfd086a07176afe7c76c2c61&pii=0021869373900926&tid=spdf-987313b5-1422-4f5b-a9e4-054d8aa1e05a&sid=e191fc0874983346178b6c043a894d10dbc8gxrqa&type=client