Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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

share|improve this question
add comment

1 Answer 1

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.

share|improve this answer
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
add comment

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.