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Given a finite group $G$, Is there any known algorithm which gives a random subgroup of $G$?

Olexandr Konovalov
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I_wil_break_you as
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    I think that is basically the same question as: is there a known algorithm which provides all subgroups of $G$? – drhab Dec 28 '18 at 11:49
  • What do you mean by a random subgroup? I could pick a random collection of elements and choose the subgroup they generate. Or do you want each subgroup to have an equal chance of being chosen? And what do you want to do with the trivial subgroup and the whole group (for some groups these are the only subgroups)? – Mark Bennet Dec 28 '18 at 12:07
  • Each subgroup have an equal chance of being chosen – I_wil_break_you as Dec 28 '18 at 12:16

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Choose a random element as cyclic subgroup generator, generate the subgroup from that element, and you already have a random cyclic subgroup.

Angel Moreno
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  • Not all subgroups are cyclic though meaning not every subgroup has an equal chance of being choosen. – Alex S Dec 29 '18 at 06:23
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Unless you have more information about the group like a Cayley table, or some information about the structure of the group, they the answer is obviously no because the group could literally be anything. If you do have this information available, then as drhab mentioned you could write a program that exhaustively searches for all the subgroups of $G$. You can save computation by using Lagrange's theorem to only check subsets of $G$ that divide the order of $|G|$ (so if $G$ is prime, you're in luck my friend!). I'm sure there are other ways you could also improve efficiency too, but that's the first one that comes to mind. After that's done, just label the subgroups from 1 to however many subgroups there are and choose a random number in that range.

Alex S
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  • Also, see https://math.stackexchange.com/questions/2082441/how-do-i-find-all-all-the-subgroups-of-a-group – Alex S Dec 28 '18 at 14:30