Timeline for Direct product of hopfian groups
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 6, 2017 at 20:32 | comment | added | Shijie Gu | Right. I think the best result Hirshon got is $A$ is strong hopfian and $B$ has finitely many normal subgroups. | |
| Dec 6, 2017 at 12:47 | comment | added | YCor | @JeremyRickard thank you! Corner's article is not cited in Hirshon's 1971 paper given by the OP, but it's indeed cited in Hirshon's 2002 survey (cited p4 - number 368) "Misbehaved direct products" mentioned in my previous comment. By the way the constructed hopfian $A$ with non-hopfian $A^2$ is abelian. | |
| Dec 6, 2017 at 10:35 | comment | added | Jeremy Rickard | @YCor A.L.S. Corner, in "Three examples of hopficity in torsion-free abelian groups" (Acta Mathematica Academiae Scientiarum Hungarica 16, pp.303-310, 1965) gives examples of two hopfian groups whose direct product is non-hopfian, and even of a hopfian group $A$ with $A\times A$ non-hopfian. | |
| Dec 5, 2017 at 23:18 | comment | added | YCor | Certainly this "finitely many normal subgroups" is an overkill and should be weakened. The most optimistic statement would be that $A,B$ hopfian implies $A\times B$ hopfian (this would be an if-and-only-if). Unfortunately it's not mentioned, and I can't find any reference in Hirshon's survey "sciencedirect.com/science/article/pii/…" (where he addresses as open problem whether hopfian is stable under taking direct product with $\mathbf{Z}$) | |
| Dec 5, 2017 at 23:14 | comment | added | YCor | OK the paper indeed starts with "Throughout the paper $A$ is a hopfian group and $B$ is (unless we specify otherwise) a group with finitely many normal subgroups". Theorem 12 says "If $B$ is a perfect group then $A\times B$ is hopfian. It is not "specified otherwise". So indeed $B$ is assumed to have finitely many normal subgroups. (This answer the question before it was edited.) | |
| Dec 5, 2017 at 23:12 | comment | added | YCor | @ARG $B$ hopfian is clearly a necessary condition for $A\times B$ to be hopfian. (And obviously a group with finitely many normal subgroups is hopfian) | |
| Dec 5, 2017 at 19:24 | comment | added | Shijie Gu | @Ycor Thanks for providing the link. I think $B$ doesn't have to be assumed to be hopfian. But the hidden hypothesis "finitely many normal subgroups" is needed in Hirshon's proof. | |
| Dec 5, 2017 at 19:23 | history | edited | Shijie Gu | CC BY-SA 3.0 |
added 88 characters in body
|
| Dec 5, 2017 at 9:22 | comment | added | YCor | It's impossible to understand your question without looking at the paper; I just did. It's indeed not very clear what are the assumptions: the statement of the theorem is "If $B$ is a perfect group then $A\times B$ is hopfian.". The abstract summarizes the results. It seems that the only standing assumptions are that $A$ and $B$ are hopfian (and, in this precise theorem, that $B$ is perfect). | |
| Dec 5, 2017 at 9:15 | comment | added | YCor | projecteuclid.org/euclid.afm/1485894451 | |
| Dec 5, 2017 at 9:14 | comment | added | YCor | This is a weird question... are you asking whether every perfect group has finitely any normal subgroups? or whether Hirshon skipped a hypothesis (in which case you should elaborate) | |
| Dec 5, 2017 at 0:36 | history | asked | Shijie Gu | CC BY-SA 3.0 |