## 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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