Timeline for If $A$, $B$ are abelian groups such that $\mathrm{Hom}(A, G) \cong \mathrm{Hom}(B, G)$ for all abelian groups $G$, must $A$ and $B$ be isomorphic?
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Mar 5, 2023 at 10:04 | comment | added | Peter LeFanu Lumsdaine | The paper of Sebel’din is available in full text here: А. М. Сбельдин, Группы гомоморфизмов вполне разложимых абелевых групп без кручения, Известия высших учебных заведений Математика, 1973, номер 7, 77–84 | |
| Mar 5, 2023 at 9:39 | comment | added | Jeremy Rickard | @EmilJeřábek True. Although that can easily be fixed by taking $\operatorname{Div}(G)$ to be the largest divisible subgroup of $G$ rather than the group of divisible elements. | |
| Mar 5, 2023 at 9:14 | comment | added | Emil Jeřábek | I should have also pointed out that if $G$ has torsion, then $\mathrm{Div}(G)$ may not be divisible. E.g., if $G$ is a $p$-group, then $\mathrm{Div}(G)$ is the Ulm subgroup $U^1(G)$, which is not divisible if $G$ has Ulm length at least $2$. | |
| Mar 5, 2023 at 8:48 | comment | added | Jeremy Rickard | @CarlosEsparza As Emil says, $\operatorname{Hom}(\mathbb{Q},G)\not\cong\operatorname{Div}(G)$ if $\operatorname{Div}(G)$ has torsion, since $\operatorname{Hom}(\mathbb{Q},G)$ is always a vector space over $\mathbb{Q}$. But the important point here is that if $\operatorname{Div}(G)$ has torsion then $\operatorname{Hom}(\mathbb{Q},G)$ is an infinite dimensional vector space. | |
| Mar 5, 2023 at 8:26 | comment | added | Emil Jeřábek | @CarlosEsparza I don’t think Hom(Q,G) is necessarily Div(G) if G is not torsion-free. Consider that e.g., every automorphism of $G=\mathbb Z_{p^\infty}$ lifts to a homomorphism $\mathbb Q\to\mathbb Z_{p^\infty}$. | |
| Mar 5, 2023 at 1:36 | comment | added | Carlos Esparza | Is this (spoiler alert) the right idea? | |
| Mar 5, 2023 at 0:55 | comment | added | LSpice | @EmilJeřábek, you missed a golden opportunity to say "Oh, ISHTOT". | |
| Mar 4, 2023 at 19:41 | comment | added | Emil Jeřábek | Oh, I see. Thank you. | |
| Mar 4, 2023 at 19:39 | comment | added | Jeremy Rickard | @EmilJeřábek If it's an Indian spirit, then that makes it much better! I meant "I Should Have Thought Of That". | |
| Mar 4, 2023 at 19:38 | comment | added | Emil Jeřábek | Google tells me Ishtot is some sort of Indian spirit. It does not know it as an acronym. Could you clarify? | |
| Mar 4, 2023 at 19:18 | history | answered | Jeremy Rickard | CC BY-SA 4.0 |