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How can I get sharp bounds for $gnu(75600)$, the number of groups of order $75600$.

I tried to determine the number of groups of order $15120$ to get a reasonable lower bound, but I quit GAP after some hours, noticing that there still was a long way to finish the calculation.

I determined $gnu(2160)=3429$ , $gnu(3024)=4635$ and $gnu(5040)=4539$, so a lower bound of $gnu(75600)$ is $gnu(3024)\cdot gnu(25)=9270$.

Can anyone give better bounds, or even the actual value ?

ahulpke
  • 20,399
Peter
  • 86,576

1 Answers1

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There are (usual method -- calculation with GAP) 22758 groups of order 15120. The calculation took about 2 days. I don't see a fundamental obstacle to running order 75600, but that might take a week or two.

ahulpke
  • 20,399
  • Did you check that the groups are actually non-isomorphic ? – Peter Apr 19 '16 at 15:57
  • This is the result of ConstructAllGroups which already uses some isomorphism tests, I did not try to determine distinguishing invariants or ran explicit isomorphism tests on the result. – ahulpke Apr 19 '16 at 20:13
  • For me, r:=ConstructAllGroups(15120) took 72hrs, the resulting list has length 22758 and ForAll(r,IsGroup) returns true meaning that the result has no lists of groups that the algorithm did not not manage to distinguish (otherwise, the list r would have elements which are lists themselves). – Olexandr Konovalov Apr 20 '16 at 22:19
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    @ahulpke: I've added an entry for gnu(15120) at https://github.com/alex-konovalov/gnu/blob/master/gnudata.g. Thinking of crowdsourcing the database of gnu(n). – Olexandr Konovalov Apr 20 '16 at 22:30
  • @AlexanderKonovalov Thank you for cross-checking the result. It would be very nice if a database of the known gnu's would be available. – Peter Apr 21 '16 at 16:34
  • For integer factorizations, such a database exists : http://factordb.com/ – Peter Apr 21 '16 at 16:36
  • @Peter: you can now submit individual Gnu(n) as new issues here. See also README. Maybe you can start with walking through your questions on Gnu(n) on this site (in this case,include their URL in the issues). – Olexandr Konovalov Apr 21 '16 at 19:49
  • Where can I see the known values and the "holes" ? – Peter Apr 21 '16 at 21:25
  • Use "@" to ping - I am not seeing your replies otherwise, and there is no guarantee that I am revisiting this page. For known values, the README mentions gnudata file. Holes - those not covered by Small Groups Library and sglppow and those beyond being reasonably computable with cubefree. – Olexandr Konovalov Apr 21 '16 at 21:45
  • Hi @Peter - please have a look at https://github.com/alex-konovalov/gnu/blob/master/WISHLIST.md – Olexandr Konovalov Apr 21 '16 at 23:33
  • @AlexanderKonovalov I would like to establish a full database availabe for everyone (without registration and without the need of a download), but I still have doubts of the reliability of the values. I remember I asked for a special gnu, two different method showed the same value, but it was wrong. That might mean that GAP still has some bugs or other issues. What do you mean. Which of the values $gnu(1)-gnu(2000)$ are really reliable ? – Peter Apr 22 '16 at 19:54
  • @AlexanderKonovalov If I could be sure that constructallgroups is absolutely reliable, I could calculate most of the values myself, unless the annyoing breakdown due to memory lack prevents this. I would suggest that everyone who found a new gnu can submit it and after it has been checked, it should be added to the database, like in the factordb-site. I see no reason why the gnu's should keep secret, I doubt that someone uses them to encrypt messages :) – Peter Apr 22 '16 at 19:57
  • Yes, crowdsourcing this collection is exactly my intention. I've submitted three entries myself today to show examples how this would work. There is no need for the user to register, and one could use any SCSCP client to connect to it without downloading the data. – Olexandr Konovalov Apr 22 '16 at 21:08
  • I think you mean the typo in the formula which was fixed in new SglPPow: http://math.stackexchange.com/q/619365/. That was not related to ConstructAllGroups. The Small Group Library for n in [1..2000] seem stable and is part of both GAP and Magma. Random checks confirm numbers. Full check would make a good answer to http://math.stackexchange.com/q/1607517/. – Olexandr Konovalov Apr 22 '16 at 21:13
  • @AlexanderKonovalov I tried to calculate $gnu(3888)$ . After several hours I got a list of $20,051$ groups and $7$ pairs of groups. Can you verify the final result that I got : $gnu(3888)=20,065$ ? – Peter Apr 27 '16 at 21:29
  • @Peter: thanks. How pairs are counted in the number 20065? If you were unable to split them or show their isomorphism, it may be too early to claim gnu(3888). – Olexandr Konovalov Apr 27 '16 at 21:35
  • @Peter: because http://www.gap-system.org/Manuals/pkg/grpconst/htm/CHAP003.htm says "However, in some cases the output might contain lists of groups as well. The groups is such a list could not be proved to be pairwise non-isomorphic by the algorithm, although this is likely to be the case, see Section Verifying non-isomorphism for further details." – Olexandr Konovalov Apr 27 '16 at 21:41
  • then http://www.gap-system.org/Manuals/pkg/grpconst/htm/CHAP004.htm#SECT004 says "If DistinguishGroups fails to split up the input list completely, then a user might use the general purpose function IsomorphismGroups to prove the non-isomorphism between the remaining groups. However, this might be a time consuming computation." and that's what is necessary to do – Olexandr Konovalov Apr 27 '16 at 21:42
  • I used the distinguishedgroups-function and set the boolean variable to "true" and the result was "true". So, probably, the $14$ groups were non-isomorphic. In this case, the result came immediately, but I have a small amount of doubt, whether the result is actually true. – Peter Apr 27 '16 at 21:46
  • Documentation for DistinguishGroups says that it "It returns a similar list, where the sublists contained in list are split up". In this case, it does not change the structure of the input list, so it's not enough. I've checked that they are non-isomorphic using IsomorphismGroups instead - see https://github.com/alex-konovalov/gnu/commit/ec2f9d18b98d1907343f3aeb2dcaf02c2f968640 – Olexandr Konovalov May 02 '16 at 12:04
  • @ahulpke: instead of a week or two, it still runs for me for 56 days already in GAP 4.8.3: https://github.com/alex-konovalov/gnu/issues/26 – Olexandr Konovalov Jun 21 '16 at 09:57