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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?

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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.)

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