## Group Theory Problems

Mike Pawliuk and I got talking about elementary group theory problems today. I wanted to record one of my favourites. I heard this one from Lucy Kadets, who heard it from Yuri Burda. I’m not sure if he is the original author or not.

Let denote the symmetric group on elements. We say is *a point-fixing subgroup* if there is a such that for all . We say *has fixed points* if for each there is such that .

Exercise 1Is every that has fixed points a point-fixing subgroup?

Mike also asked some charming questions:

Exercise 2Is there a countable group into which every finite group embeds isomorphically?

Exercise 3Is there a group with more than a million elements, each element of which is self-inverse?

Yura Burdasaid, on 2013/04/08 at 05:13Just wanted to comment that exercise 1 is related (via Artin symbol/Chebotarev density) to the following lovely number-theoretic question: are there polynomials with integer coefficients that have a root modulo any prime, but not over the rationals?

pgadeysaid, on 2013/04/09 at 12:39Ooh! That is a really lovely question.

I don’t know enough number theory to see immediately how that should work, but I’ll think about it. Thanks for pointing out where the problem came from.