Timeline for Which finite groups can be characterized by their subgroup orders?
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
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| Apr 25, 2014 at 7:19 | comment | added | Tobias Kildetoft | Generalizing the comment of @GerryMyerson this holds for any group whose order is a cyclic number (and these can be recognized from the set of subgroup orders by these all being square free and the product of the set of their prime divisors being a cyclic number). | |
| Apr 24, 2014 at 23:35 | history | edited | Marius Tarnauceanu | CC BY-SA 3.0 |
added 209 characters in body
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| Apr 24, 2014 at 23:20 | comment | added | Marius Tarnauceanu | Inspired by the above comment by P. Vanchinathan, I've added an additional question. | |
| Apr 24, 2014 at 22:32 | vote | accept | Marius Tarnauceanu | ||
| Apr 24, 2014 at 22:32 | vote | accept | Marius Tarnauceanu | ||
| Apr 24, 2014 at 22:32 | |||||
| Apr 24, 2014 at 22:27 | comment | added | Marius Tarnauceanu | $\pi_s(\mathbb{Z}_4)=\pi_s(\mathbb{Z}_2\times\mathbb{Z}_2)=\{1,2,4\}$, i.e. these groups cannot be characterized by their subgroup orders. On the other hand, we can replace the set of subgroup orders with the multiset of subgroup orders, and in this case both $\mathbb{Z}_4$ and $\mathbb{Z}_2\times\mathbb{Z}_2$ are uniquely determined. | |
| Apr 24, 2014 at 13:53 | answer | added | Derek Holt | timeline score: 15 | |
| Apr 24, 2014 at 13:23 | comment | added | P Vanchinathan | How do you handle it when there are many suggroups of the same order. If you discount multiplicity the two groups of order 4 will have the same 'horoscope'. | |
| Apr 24, 2014 at 12:39 | comment | added | Gerry Myerson | The property also holds for many non-simple groups, e.g., groups of order $pq$ with $p<q$ primes and $q-1$ not a multiple of $p$. | |
| Apr 24, 2014 at 11:02 | comment | added | Derek Holt | A computer calculation shows that the simple groups of orders up to $2000$ are all characterized by their subgroup orders. It seems to be a reasonable conjecture that this might be true in general, but it is hard to think how you might go about proving it. | |
| Apr 24, 2014 at 9:53 | comment | added | Neil Hoffman | You might consider checking your question against the data here: madore.org/~david/math/simplegroups.html | |
| Apr 24, 2014 at 9:36 | history | asked | Marius Tarnauceanu | CC BY-SA 3.0 |