What is an example of an infinite group with a composition series and infinitely many simple subgroups?
Example of infinite group with infinitely many simple subgroups
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$\begingroup$
abstract-algebra
finite-groups
1 Answers
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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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1first doesn't quite work (composition series are generally finite), but the second is fine. – 2012-11-12
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0Why 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?) – 2012-11-12
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1@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. – 2012-11-12
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0I see - I think the part I was missing was that $A_{\text{finitary}}$ is also simple. – 2012-11-12