Timeline for Which group does not satisfy the Tits alternative?
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| Sep 15, 2010 at 2:02 | comment | added | Pete L. Clark | @MB: Thanks for the help. For some reason I was vaguely doubtful of 1., but now that you assert it I see how to prove it (a subgroup of a solvable group is solvable!). | |
| Sep 14, 2010 at 20:09 | comment | added | Maxime Bourrigan | What I meant is : 1. a subgroup of a virtually solvable group is virtually solvable. So F cannot be virtually solvable if F' isn't. 2. An infinite simple group has no finite-index subgroup at all (it is easy to show that every finite-index subgroup contains a finite-index normal subgroup). So, if it were virtually solvable, it would be solvable, but that's absurd. | |
| Sep 14, 2010 at 18:51 | comment | added | Pete L. Clark | I'm a bit confused by the first parenthetical remark, which seems to suggest that simplicity of the commutator subgroup implies virtual solvability. Certainly this is not literally true -- e.g. $S_n$ for $n \geq 5$. But even supposing that the commutator subgroup $G'$ is infinite, I'm still not quite seeing it: off the top of my head, I would think that you need to know also that $G'$ has finite index in $G$. Please help... | |
| Sep 14, 2010 at 16:29 | history | edited | Maxime Bourrigan | CC BY-SA 2.5 |
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| Sep 14, 2010 at 16:18 | history | edited | Maxime Bourrigan | CC BY-SA 2.5 |
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| Sep 14, 2010 at 16:12 | history | answered | Maxime Bourrigan | CC BY-SA 2.5 |