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What is an example of an infinite group with a composition series and infinitely many simple subgroups?

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One example is the direct sum of all the finite simple groups (more precisely, pick one for each isomorphism class).

Another (perhaps less cheat-y) one is the group of permutations of $\mathbb N$, which contains all the alternating groups $A_n$ as subgroups.

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    I see - I think the part I was missing was that $A_{\text{finitary}}$ is also simple.2012-11-12