## Symmetry @ Otterbein

The image represents a molecule of chloropentacarbonylmanganese or ClMn(CO)_{5}. For the last half-hour or so, I have been playing around in the Symmetry Gallery put together by Otterbein. Lots of lovely molecules with high degrees of symmetry. It is remarkable (to me) that such things even exist in nature. I encourage you to go and check out the lovely interactive demos on the site.

## The Rotationally Distinct Ways to Label a Die

I’m giving a talk at the Canadian Math Camp this year. I’ll be showing the kids of how to count the number of ways to label a six sided die up to the rotational symmetries of the cube. Here is the handout for the talk with questions about dice labellings, the 15-puzzle, and permutation groups.

For the curious the labellings are below the cut. Please note that there are typos in the table below. Alex Fink kindly pointed them out and they will be fixed eventually. For now they are an exercise in keen observation.

## 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?

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