# Examples of profinite simple groups

The standard example of an infinite simple group is $A_\infty$, the direct limit of the alternating groups under the obvious injections.

Are there also examples of infinite simple groups arising as the inverse limits of finite groups, i. e. profinite groups?

• A bit hard to do since they'd need to map down to the limitands...? – paul garrett Apr 1 '13 at 23:27
• You're right, my question is nonsense. (facepalm) – Dominik Apr 1 '13 at 23:29

The finite groups $H_l$ (in the inverse system whose limit is a given profinite group $G$) correspond to finite index normal subgroups $N_l\triangleleft G$ - namely, $N_l=\ker(G\to H_l)$ - so $G$ is nonsimple by design.
A profinite group is simple if and only if it is a finite simple group. Indeed, if $G$ is simple and non-trivial, we can choose $g\in G$ which is not the identity. Because $G$ is Hausdorff, there is an open neighborhood $U$ of the identity with $g\notin U$. Because $G$ is profinite, there is an open normal subgroup $N$ of $G$ contained in $U$. Since $N\neq G$, $N$ is the trivial subgroup by simplicity. But then the identity is open, so every point of $G$ is open, and thus $G$ is discrete, hence finite.