LARGE countable groups

O. Kegel

A countable group \(G\) will be called small if it is finitely generated; and LARGE if it is infinitely generated and acts transitively (by conjugation) on the set of all embeddings of any small group \(X\) into \(G\). The set \(sk(G)\) of all (isomorphism types of) finitely generated subgroups \(X\) of any countable group \(G\) is also (at most) countable. Every countably infinite group \(G\) is the union of an ascending sequence \(\{G(n) \mid n\in \mathbb{N}\}\) of small subgroups of \(G\).

Theorem. Two large countable groups, \(G\) and \(H\), are isomorphic if and only if \(sk(G)=sk(H).\)

(Close connection to countable existentially closed groups.)