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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