# Example of infinite group with infinitely many simple subgroups

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

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## 1 Answer

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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first doesn't quite work (composition series are generally finite), but the second is fine. – Jack Schmidt Nov 12 '12 at 19:16
Why is the composition series in the second case finite? Also, to be clear, you mean all permuations of $\mathbb{N}$ or only the finitely supported ones? (Or does it matter?) – Jason DeVito Nov 12 '12 at 20:35
@JasonDeVito: only matters a little. both have composition series of finite length, but different lengths. Alt(finitary) <= Sym(finitary) <= Sym(all N) or so, I believe. Scott's Group Theory textbook has a nice description of composition series of symmetric groups of infinite sets. – Jack Schmidt Nov 12 '12 at 21:12
I see - I think the part I was missing was that $A_{\text{finitary}}$ is also simple. – Jason DeVito Nov 12 '12 at 21:15