Does there exist a group $G$ such that


1. for any finite $K$ there is a monomorphism $K \to G$
2. for any $H$ with property 1 there is a monomorphism $G \to H$


If yes, is it the only one?



Is there a smallest group containing all finite groups?
No. To show that it doesn't exist it is enough to produce two groups $G,H$ which contain isomorphic copies of all finite groups, but such that no group containing isomorphic copies of all finite groups embeds into both $G$ and $H$.


Let $(G\_n)$ be an enumeration of all finite groups. Let $G=\bigoplus G\_n$ be the restricted direct sum and $H={\Large\ast}\_nG\_n$ the free product.


If $K$ is a subgroup of $G$ then $K$ is locally finite, hence freely indecomposable. Hence if $K$ is also isomorphic to a subgroup of $H$, then by Kurosh's subgroup theorem, $K$ is finite. In particular, $K$ doesn't contain isomorphic copies of all finite groups.


