Timeline for Extensions isomorphic as groups but not congruent or pseudo-congruent
Current License: CC BY-SA 2.5
15 events
| when toggle format | what | by | license | comment | |
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| Aug 16, 2010 at 4:06 | comment | added | Jack Schmidt | Cool, I added your example in the answer since the comments are so long. | |
| Aug 16, 2010 at 4:05 | history | edited | Jack Schmidt | CC BY-SA 2.5 |
include vipul's example since the emphasis is on E, not G
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| Aug 15, 2010 at 22:58 | comment | added | Vipul Naik | So, to summarize: order 2^5 (and no smaller): get an example where <em>A</em> abelian, <em>G</em> class two. order 2^6 (and no smaller): get an example where <em>A</em> and <em>G</em> both abelian. order 2^7 (and no smaller): get an example where the big group is abelian as well. | |
| Aug 15, 2010 at 22:57 | comment | added | Jack Schmidt | @Vipul: yeah, the order 64 guy is very similar to your example. | |
| Aug 15, 2010 at 22:56 | comment | added | Jack Schmidt | @Vipul: cool. I also noticed the first order 64 example had G abelian, so I wrote it up in the answer. | |
| Aug 15, 2010 at 22:55 | history | edited | Jack Schmidt | CC BY-SA 2.5 |
abelian example
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| Aug 15, 2010 at 22:51 | comment | added | Vipul Naik | Actually, it turns out I already have a counterexample when the big group is abelian and of order $p^7$, I just hadn't connected it to the question since it was obtained in a different context: groupprops.subwiki.org/wiki/… | |
| Aug 15, 2010 at 22:49 | comment | added | Jack Schmidt | Also, if you drop the "central" / "trivial action" requirement, then the multiple A silliness is pretty common in all groups (the "general" group only rarely has central factors, so having multiple, non-conjugate central factors is also rare). This problem was one of the reason the Holt–Plesken perfect group project tapered off at the point it did: deciding queer isomorphisms was getting too hard. However, if one was happy only finding the groups with small lower central factors (of the Fitting series of the solvable radical), then one could go much further. | |
| Aug 15, 2010 at 22:46 | comment | added | Jack Schmidt | If you want to generate p-groups systematically by their central series, then the "p-group generation algorithm" (implemented nicely in ANU pq for GAP and magma) is the way to go. This silliness with multiple copies of A goes away if you can "name" A, say as the last term of the lower p-central series. | |
| Aug 15, 2010 at 22:44 | comment | added | Jack Schmidt | @Vipul: Thanks for the typo catch (search was coded up nicely, examining the group was done by hand). I wouldn't be too surprised either way for G abelian: on the one hand you just need some room to work in order for different Aut(G) orbits of Proj(H^2(G,A)) to be abstractly isomorphic, but on the other hand Aut(G) for G elementary abelian is really pretty transitive, so maybe there are not enough orbits. But for G nilpotent of class 2, I would expect a ton of examples. 11% of order 64 have examples. | |
| Aug 15, 2010 at 22:40 | history | edited | Jack Schmidt | CC BY-SA 2.5 |
correct typo
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| Aug 15, 2010 at 21:42 | comment | added | Vipul Naik | Also, do you think an example of this sort would exist for <em>G</em> abelian as well? | |
| Aug 15, 2010 at 21:35 | comment | added | Vipul Naik | I checked this example using GAP and GAP seems to say that the two quotients should be SmallGroup(16,11) each (which is $D_8 \times \mathbb{Z}_2$) rather than SmallGroup(16,3). Perhaps you had some other group in mind? Anyway, this one works. | |
| Aug 15, 2010 at 21:27 | vote | accept | Vipul Naik | ||
| Aug 15, 2010 at 15:04 | history | answered | Jack Schmidt | CC BY-SA 2.5 |