Quoting English Wikipedia: 'a theorem of Graham Higman states that a finitely generated group has a recursive presentation if and only if it can be embedded in a finitely presented group'.


I'd like to ask if there is any nice criterion to characterize groups that can be embedded just in a finitely generated group?



Characterization of subgroups of finitely generated groups
A group embeds in a finitely (even 2-) generated group iff it is countable. It was proved by Higman, B. Neumann and H. Neumann (Higman, Graham; Neumann, B. H.; Neumann, Hanna
Embedding theorems for groups. 
J. London Math. Soc. 24, (1949). 247–254.)


