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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