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I was studying the following problem :

Groups of order $n^2$ that have no subgroup of order $n$

Is there any other divisor than 24 of $24^2$ such that SmallGroup(576,8661) has no subgroup of that order.

1 Answers 1

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This GAP-output :

gap> Collected(List(AllSubgroups(SmallGroup(576,8661)),Order));
[ [ 1, 1 ], [ 2, 63 ], [ 3, 64 ], [ 4, 651 ], [ 8, 1395 ], [ 9, 64 ],
  [ 12, 336 ], [ 16, 651 ], [ 32, 63 ], [ 48, 84 ], [ 64, 1 ], [ 192, 1 ],
  [ 576, 1 ] ]
gap> DivisorsInt(576);
[ 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 32, 36, 48, 64, 72, 96, 144, 192, 288,
  576 ]
gap>

shows that there are no subgroups of order $6$, for example. You can find out all missing orders easily using this output.

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    Thank you very much. I would like to learn GAP. I have already installed it. Please give me some references. – 2017-02-20
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    Just read the tutorials. You will find them on some directory. Or input "?groups", for example, to get immediate help. – 2017-02-20
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    It will be more than three times faster to check orders of representatives of conjugacy classes of subgroups. E.g. `s:=Set(ConjugacyClassesSubgroups(SmallGroup(576,8661)),c->Size(Representative(c)));` returns `[ 1, 2, 3, 4, 8, 9, 12, 16, 32, 48, 64, 192, 576 ]`. Now `Difference(DivisorsInt(576),s);` returns `[ 6, 18, 24, 36, 72, 96, 144, 288 ]`. – 2017-02-20
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    In addition to documentation supplied with GAP (and also available online at https://www.gap-system.org/Doc/manuals.html), there is also a Software Carpentry lesson on GAP: http://alex-konovalov.github.io/gap-lesson/. I taught it at two workshops, but it also should be accessible for self-study. – 2017-02-20
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    @AlexanderKonovalov Hi, anything new about the group-project ? – 2017-02-20
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    @Peter hi, good to hear from you! I've stopped adding new entries, since I want to upgrade database first, but was diverted by other things - maybe in Spring. But the SCSCP server is up and running so calling `GnuFromServer` etc. works as described at https://github.com/alex-konovalov/gnu – 2017-02-20
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    @AlexanderKonovalov Did you continue to search a prime factor of my number ? – 2017-02-20
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    How to find order of all normal subgroup of a particular group in GAP? – 2017-02-22
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    @SSSM The command is NormalSubgroups() – 2017-02-22