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Is there a standard name for this concept: Let $H \leq G$ be groups. Say $H$ is ?? if there is a finitely-generated group $K \leq G$ such that $H \leq K$. What should one use in place of "??"? I'm also interested in this notion up to isomorphism.

I've used "finitely subgenerated", though I have realized that "finitely supergenerated" probably makes a bit more sense. If you know a reference/situation where such a thing plays a key role, that may also be useful even if no name is given.

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  • $\begingroup$ If $H$ were also finitely-generated, then “finitely subgenerated” would mesh well with current similar terms, like “subnormal.” You might want to take a tour through groupprops to see if you can find any terms that handle a similar condition, but I wasn’t able to. $\endgroup$ May 22, 2019 at 13:59

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I'm quite unaware of any such terminology.

As a point of comparison, there is even an important theorem called Higman's Embedding Theorem which characterizes, up to isomorphism, those groups which are isomorphic to subgroups of finitely presented groups. Such groups do have a special terminology, but that terminology arises not from their embeddability into a finitely presented group but instead from the other side of the characterization in Higman's theorem --- they are precisely the recursively presented groups.

So I suggest that if you have a need for such terminology, you judiciously choose it for yourself.

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