Is every group isomorphic to an inverse limit (that is, projective limit) of perfect groups?


I guess, the answer is no. In that case: Is there a characterization of the groups that are isomorphic to inverse limits of perfect groups?



Inverse limits of perfect groups
Yes:



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> Every group is isomorphic to a filtering projective limit of a sequence of perfect groups.
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Start from a group $G$ and embed it into a perfect group $P$ (e.g., a symmetric group). 


Consider the group $H$ obtained by amalgamating countably many copies $P\_n$ of $P$ over $G$. Then $H$ is perfect. Let $H\_n$ be the subgroup generated by $\bigcup\_{k\ge n}P\_k$; it is isomorphic to $H$ and also perfect. The decreasing intersection of the $H\_n$ is reduced to $G$. Hence $G$ is an filtering projective limit of perfect groups.


(This can sound a bit surprising, since one mostly works with projective limits of surjections, in which case many groups cannot be obtained.)


