Since I wrote one of them, I've always been interested in papers that show that "almost all groups do or have some property..." (for the paper Cary Huffman and I wrote, which is available from my home page, the result was that the ring of invariant polynomials was as messy as it could possibly be) . Google will show you many similar papers. For Dan's question, there will be two answers, just as there are for subgraphs of a graph, depending on whether the (nodes or group elements) are labeled or unlabeled Both versions seem interesting. Best regards Neil Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com On Fri, May 4, 2018 at 7:00 PM, Dan Asimov <dasimov@earthlink.net> wrote:
The problem was not what I wrote way below: Obviously with a finite number of groups of order n, one can easily pick one "at random".
But how about the subgroups? Are we picking one from *all* subgroups, including all those conjugate to one subgroup?
That method will tend to decrease the probability of a normal subgroup, since every *other* subgroup will get counted once for each of its conjugates but a normal subgroup has only one conjugate.
On the other hand, picking just one subgroup from each conjugacy class will increase the probability of a normal subgroup.
So: What is the probability measure being considered?
—Dan
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(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 Propp wrote: ----- for large n, if you pick a group G of order n uniformly at random -----
I wonder how to define a probability on groups of order n.
And in particular: If there are several ways to do this, do they result in the same probabilities?
—Dan
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