Timeline for Uncountable counterexamples in algebra
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 12, 2021 at 18:14 | comment | added | Jeremy Rickard | @MartinBrandenburg It's true for singular cardinals, such as $\aleph_{\omega}$, but false for $\aleph_n$ ($n$ finite and greater than $0$). I don't think it's known for precisely which $\lambda$ it is true. | |
| May 11, 2021 at 18:41 | comment | added | Martin Brandenburg | This is not too surprising, since "countable" is the next thing after "finite". What about the obvious generalization to cardinal numbers $\lambda$? If $G$ is an abelian group with $\lambda$ elements and every subgroup of rank $< \lambda$ is $\mathbb{Z}$-free, does it follow that $G$ is free? Or is $\mathbb{Z}^{\mathbb{N}}$ with $\lambda=2^{\aleph_0}$ a counterexample (perhaps assuming $\mathrm{CH}$)? Is it true for some cardinals $> \aleph_0$ at least? | |
| May 9, 2021 at 20:56 | history | made wiki | Post Made Community Wiki by Stefan Kohl♦ | ||
| May 9, 2021 at 20:26 | comment | added | Nik Weaver | Excellent, just the sort of thing I was looking for. | |
| May 9, 2021 at 20:09 | history | answered | YCor | CC BY-SA 4.0 |