It's four days before All Hallow's Day, so I guess that makes today Hallowe'en-e'en-e'en-e'en? Here's another pumpkin spice flavored Advanced Knowledge Problem of the Week to celebrate it.
Keep reading to see my formulation of the solution!
Did you come up with another Cayley table for the quotient groups, and what representative elements did you use? Do you want to see more chalkboard drawings in the Problems of the Week? Let us know in the comments section, and have a happy Halloween!
Hey I recognize this group H! It's the quaternions Q8. We can observe that it's a group of order 8 that is not abelian, so it's either D8 (group of symmetries of the square) or Q8. It can't be D8 because there is only one element of order two (jack-o-lantern) and a square has more than one symmetry axis. So it's Q8 (or H for Hamilton) and jack-o is therefore (-1). Anyway:
ReplyDeleteReading the line for Jack-o-L gives you the pairings for the opposites (eg "skull" is the opposite of "ghost" because Jack-o-L times skull=ghost). Now you only keep one drawing from each pairing and draw the Cayley table with just four elements. You see that every element's square is a pumpkin. This is the Klein four-group (symmetries of the rectangle).