Parker Glynn-Adey

Group Theory Problems

Posted in Math by pgadey on 2013/03/26

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 {S_n} denote the symmetric group on {n} elements. We say {H \leq S_n} is a point-fixing subgroup if there is a {1 \leq k \leq n} such that {h(k) = k} for all {h \in H}. We say {H} has fixed points if for each {h \in H} there is {1 \leq k_h \leq n} such that {h(k_h) = k_h}.

Exercise 1 Is every {H \leq S_n} that has fixed points a point-fixing subgroup?

Mike also asked some charming questions:

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

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

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2 Responses

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  1. Yura Burda said, on 2013/04/08 at 05:13

    Just 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?

    • pgadey said, on 2013/04/09 at 12:39

      Ooh! 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.

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