TITLE: Fraïssé limits and groups
QUESTION [14 upvotes]: I was recently reading up on Fraïssé limits in Hodges' "A shorter model theory." I was trying to think of some examples and wanted to see if I could take the Fraïssé limit on the category of finite groups. It is clear that this category has the hereditary and joint embedding property. However, I am having trouble showing that this category has the amalgamation property, by which I mean that if there are finite groups $G, H, K$ such that there are embeddings $\varphi: G\hookrightarrow H$ and $\psi: G\hookrightarrow K$, then there exists a finite group $J$ and embeddings $\vartheta: H\hookrightarrow J$ and $\eta: K\hookrightarrow J$ such that $\vartheta\circ \varphi = \eta\circ \psi$. 
One simplification is that we can, by taking isomorphic copies if necessary, assume that $H\cap K = G$, and that $\varphi $ and $\psi$ are just inclusion maps. I am not even sure what group $J$ should be. Any help would be appreciated.

REPLY [2 votes]: The Fraisse limit of the class of finite groups has been studied by Hall [Hall, P. Some constructions for locally finite groups. J. London Math. Soc. 34 (1959) 305--319. MR162845] and is therefore called Hall's universal group.