[math-fun] Normal subgroups
Has anyone proved that normally subgroups aren't normal? (That is, for large n, if you pick a group G of order n uniformly at random, and then pick a subgroup H of G uniformly at random, the probability that H is a normal subgroup of G goes to 0 as n goes to infinity.) Jim
This is just not intuitively obvious to me. There are very common classes of integers, so that for any n in the class, the chance of a random subgroup of a random group of order n being normal is quite high. For example, in the class of prime numbers, the only groups of order n are cyclic, and the only (two) subgroups are normal. Now, the chance of picking a prime n gets small as n gets large, but it doesn't get small very fast. It seems plausible to me that there might be classes of nonzero density, that have a large proportion of normal subgroups. On Fri, May 4, 2018 at 4:07 PM, James Propp <jamespropp@gmail.com> wrote:
Has anyone proved that normally subgroups aren't normal?
(That is, for large n, if you pick a group G of order n uniformly at random, and then pick a subgroup H of G uniformly at random, the probability that H is a normal subgroup of G goes to 0 as n goes to infinity.)
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My intuition is that (a) most groups have at least one non-normal subgroup, and (b) where there's one non-normal subgroup, there are lots of them (since conjugations via elements of G map them all over the place). But that's just my gut feeling. Jim On Fri, May 4, 2018 at 4:38 PM, Allan Wechsler <acwacw@gmail.com> wrote:
This is just not intuitively obvious to me. There are very common classes of integers, so that for any n in the class, the chance of a random subgroup of a random group of order n being normal is quite high. For example, in the class of prime numbers, the only groups of order n are cyclic, and the only (two) subgroups are normal. Now, the chance of picking a prime n gets small as n gets large, but it doesn't get small very fast. It seems plausible to me that there might be classes of nonzero density, that have a large proportion of normal subgroups.
On Fri, May 4, 2018 at 4:07 PM, James Propp <jamespropp@gmail.com> wrote:
Has anyone proved that normally subgroups aren't normal?
(That is, for large n, if you pick a group G of order n uniformly at random, and then pick a subgroup H of G uniformly at random, the probability that H is a normal subgroup of G goes to 0 as n goes to infinity.)
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Perhaps most groups are of order 2^N? --Rich ------ Quoting James Propp <jamespropp@gmail.com>:
My intuition is that (a) most groups have at least one non-normal subgroup, and (b) where there's one non-normal subgroup, there are lots of them (since conjugations via elements of G map them all over the place). But that's just my gut feeling.
Jim
On Fri, May 4, 2018 at 4:38 PM, Allan Wechsler <acwacw@gmail.com> wrote:
This is just not intuitively obvious to me. There are very common classes of integers, so that for any n in the class, the chance of a random subgroup of a random group of order n being normal is quite high. For example, in the class of prime numbers, the only groups of order n are cyclic, and the only (two) subgroups are normal. Now, the chance of picking a prime n gets small as n gets large, but it doesn't get small very fast. It seems plausible to me that there might be classes of nonzero density, that have a large proportion of normal subgroups.
On Fri, May 4, 2018 at 4:07 PM, James Propp <jamespropp@gmail.com> wrote:
Has anyone proved that normally subgroups aren't normal?
(That is, for large n, if you pick a group G of order n uniformly at random, and then pick a subgroup H of G uniformly at random, the probability that H is a normal subgroup of G goes to 0 as n goes to infinity.)
Jim _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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Evidence using GAP N:=Number of groups of order at most 2000 = 49910529484 k:=Number of groups of order 2^10 = 49487365422 k/N = .99152154732929340606... So with probability >.99 a group of order at most 2000 has order 2^10. So who knows about subgroups of 2-groups? On Fri, May 4, 2018 at 5:20 PM, <rcs@xmission.com> wrote:
Perhaps most groups are of order 2^N? --Rich
------ Quoting James Propp <jamespropp@gmail.com>:
My intuition is that (a) most groups have at least one non-normal subgroup, and (b) where there's one non-normal subgroup, there are lots of them (since conjugations via elements of G map them all over the place). But that's just my gut feeling.
Jim
On Fri, May 4, 2018 at 4:38 PM, Allan Wechsler <acwacw@gmail.com> wrote:
This is just not intuitively obvious to me. There are very common classes
of integers, so that for any n in the class, the chance of a random subgroup of a random group of order n being normal is quite high. For example, in the class of prime numbers, the only groups of order n are cyclic, and the only (two) subgroups are normal. Now, the chance of picking a prime n gets small as n gets large, but it doesn't get small very fast. It seems plausible to me that there might be classes of nonzero density, that have a large proportion of normal subgroups.
On Fri, May 4, 2018 at 4:07 PM, James Propp <jamespropp@gmail.com> wrote:
Has anyone proved that normally subgroups aren't normal?
(That is, for large n, if you pick a group G of order n uniformly at random, and then pick a subgroup H of G uniformly at random, the probability that H is a normal subgroup of G goes to 0 as n goes to infinity.)
Jim _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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On Fri, May 4, 2018 at 4:07 PM, James Propp <jamespropp@gmail.com> wrote:
Has anyone proved that normally subgroups aren't normal?
(That is, for large n, if you pick a group G of order n uniformly at random, and then pick a subgroup H of G uniformly at random, the probability that H is a normal subgroup of G goes to 0 as n goes to infinity.)
As Allan has pointed out, if n is prime, the probability that a random subgroup of a random group of order n is normal is 1. A series that has infinitely many 1's in it cannot converge to 0, so that's enough to show your conjecture is false. You can patch it up by saying "for each n, choose a group of order <= n at random and then a subgroup of that at random. The probability that this subgroup is normal might go to 0. but I'm not at all sure. if n = pq,then if p is congruent to 1 mod q or vice versa, there is a non-abelian group otherwise, there is only the abelian one. So the chances for order pq are considerably more than .5 that the group is normal. How about groups of order pqr? How many nonabelian groups are there, and how many normal and non-normal subgroups do they have? Andy
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-- Andy.Latto@pobox.com
participants (5)
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Allan Wechsler -
Andy Latto -
James Propp -
rcs＠xmission.com -
W. Edwin Clark