Timeline for are there finite nonabelian characteristic quotients $G$ of $F_2$ inducing a surjection $Aut(F_2)\twoheadrightarrow Aut(G)$?
Current License: CC BY-SA 3.0
12 events
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| Sep 29, 2016 at 13:22 | vote | accept | stupid_question_bot | ||
| Sep 29, 2016 at 13:21 | history | edited | stupid_question_bot | CC BY-SA 3.0 |
added 107 characters in body
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| Sep 28, 2016 at 21:28 | comment | added | HJRW | @ArturoMagidin, of course, thanks for the reminder. | |
| Sep 27, 2016 at 22:00 | comment | added | Arturo Magidin | @HJRW: There are no varieties in which all groups are finite, but there certainly are varieties in which the finitely generated relatively free groups are finite (for example, the Burnside varieties of exponents $3$, $4$, and $6$ are locally finite, hence the finitely generated relatively free groups are finite); likewise, the relatively free groups of finite rank in $\mathfrak{N}_2\cap\mathfrak{B}_n$ are all finite (as are, trivially, the finitely generated relatively free groups in the varieties of abelian groups of exponent $n>1$). | |
| Sep 27, 2016 at 21:57 | answer | added | Luc Guyot | timeline score: 7 | |
| Sep 27, 2016 at 20:30 | comment | added | Derek Holt | I think that $C_n \times C_n$ works only for small $n$ (probably $n=2,3,4,6$) because all automorphisms induced by elements of ${\rm Aut}(F)$ have determinant $\pm 1$. | |
| Sep 27, 2016 at 20:04 | comment | added | HJRW | @ArturoMagidin, iirc, there are no varieties of finite groups (and in particular no varieties in which the relatively free groups are finite). | |
| Sep 27, 2016 at 19:26 | answer | added | Derek Holt | timeline score: 6 | |
| Sep 27, 2016 at 18:16 | comment | added | Arturo Magidin | Would $K$ verbal work? The quotient would be the relatively free group of rank $2$ in the relevant variety of groups (your example being the case of $K$ the verbal subgroup generated by $xyx^{-1}y^{-1}$ and $x^n$). (You'd want a variety where the relatively free group is finite, of course...) | |
| Sep 27, 2016 at 17:32 | history | undeleted | stupid_question_bot | ||
| Sep 27, 2016 at 17:30 | history | deleted | stupid_question_bot | via Vote | |
| Sep 27, 2016 at 17:30 | history | asked | stupid_question_bot | CC BY-SA 3.0 |