Timeline for Groups with "just not" a property
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Feb 16, 2021 at 10:46 | comment | added | YCor | Oops, sorry: sure, you're right, a just-infinite group with no nontrivial finite quotient is indeed simple. But Higman's argument is not really the same, since it doesn't make use of finite presentation (or, if you like, it's the old version of the argument, with $\mathcal{P}$ being the class of trivial groups). | |
| Feb 16, 2021 at 10:02 | comment | added | ARG | @Ycor I wrote that if you start with a group $G$ which has no finite quotients (like Higman's group), and look at the "just infinite" quotient $Q$ of $G$, then $Q$ is simple (since any proper quotient of $Q$ is finite, and is a quotient of $G$ it must be trivial). This could be a earlier instance of this trick. Thanks for the reference to McCarthy! | |
| Feb 16, 2021 at 9:53 | comment | added | YCor | With $\mathcal{P}=$"finite" what you obtain is a "just infinite group" (this is not the same as simple). It appears in two papers by Donald McCarthy (1968, 1970). I haven't checked if it includes the observation that every infinite f.g. group has a just infinite quotient. | |
| Feb 16, 2021 at 8:22 | history | edited | ARG | CC BY-SA 4.0 |
corrected mistage with minimax
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| Feb 16, 2021 at 7:13 | comment | added | ARG | @Ycor thanks for this reference. But isn't this trick (perhaps not with this streamlined argument) prior to that? For example, if I have an infinite group with no [nontrivial] finite quotients, then applying this trick with $\mathcal{P}$=finite, I get a simple group. | |
| Feb 16, 2021 at 7:08 | history | edited | ARG | CC BY-SA 4.0 |
correct first sentence (did not make sense) and corrected hypothesis
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| Feb 15, 2021 at 21:31 | comment | added | YCor | Maybe especially the Robinson-Wilson 1984 groups? I remember discussing this argument I think I read there, with Breuillard in December 2004. | |
| Feb 15, 2021 at 21:11 | history | asked | ARG | CC BY-SA 4.0 |