Timeline for Hopfian property preserved by extensions with finite kernel?
Current License: CC BY-SA 4.0
8 events
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| Mar 21, 2020 at 16:21 | history | edited | YCor | CC BY-SA 4.0 |
fixed broken link
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| Dec 3, 2016 at 20:27 | comment | added | YCor | @AndreasThom you're right. The modification by "BS." of my argument (see my answer) seems to fix the issue (in any case, $H$ will not be residually finite, because a finite-by-(residually finite) finitely generated group is always Hopfian). | |
| Mar 30, 2011 at 14:24 | comment | added | Andreas Thom | That is a very nice example. But how does one show that killing $t^{-2}$ gives a Hopfian group? I doubt that this quotient is residually finite. | |
| Mar 29, 2011 at 4:05 | comment | added | user6976 | In fact, as I was told by Yves de Cornulier, the group is not a quotient of Abels' group $A_n$ by its center but the quotient of the analog $B_n$ of group $A_n$ over $\mathbb{F}_p[t,t^{-1}]$ over a central copy of $\mathbb{F}_p[t]$ (see 5.10 in the paper). Then the factor-group is not Hopfian, its factor by the (finite) subgroup generated by $t^{-2}$ is Hopfian. If one then kills $t^{-1}$ as well, one get a non-Hopfian group again. I hope he himself will explain here the Hopfian property of these groups. | |
| Mar 28, 2011 at 21:20 | comment | added | user6976 | Take the Abels' group $A_n$ (pages 19,20). Then $A_n/Z$ is not Hopfian while there is a central cyclic subgroup there F such that $(A_n/Z)/F$ is Hopfian. | |
| Mar 28, 2011 at 19:06 | comment | added | Igor Belegradek | @Mark, could you tell on what page the group is constructed? | |
| Mar 28, 2011 at 13:36 | history | edited | user6976 | CC BY-SA 2.5 |
added 7 characters in body
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| Mar 27, 2011 at 10:53 | history | answered | user6976 | CC BY-SA 2.5 |