Question
Part A. From this problem:
Label the vertices of a square 1, 2, 3, 4 in counterclockwise order, as below. The group D4 acts on the vertices as permutations (e.g., r acts as (1234) and s acts as (24)), and thus also D4 acts as permutations on pairs of vertices: r({1,2}) = {2.3}, r({l,3}) = {2,4}, and so on. There are six pairs of vertices: P = {{1,2}, {1,3}, {1,4}, {2,3}, {2,4}, {3,4}}. Find the disjoint cycle decomposition of r and s as permutations of the six elements of P. Determine the orbits of the action of D4 on P. Compute the stabilizer of {1, 2} and the stabilizer of {1, 3}.
Explanation / Answer
et's start with the orbit containing a, first. r takes a to d, so the orbit of a contains d. by inspection, we see that the orbit of a must contain every element of the 4-cycle (a d f c): r2(a) = f r3(a) = c we don't get any new elements of the orbit of a from applying s (so now new elements from applying r or s in any combination), so the orbit containing a is {a,c,d,f}. that leaves the orbit containing b. r(b) = e, so the orbit of b contains e. s leaves {b,e} fixed, so there were are: {{a,c,d,f},{b,e}} is the set of orbits. thus this action isn't transitive, we have more than one orbit (you can think of it this way: the vertex-pairs {1,2}, {2,3}, {3,4} and {1,4} really "represent the square (sides)" the vertex-pairs {1,3} and {2,4} represent the diagonals (think of the square as having an X inside it). it should be clear that an element of D4 maps "sides to sides" and "diagonals to diagonals"). if you REALLY want to blow your mind, think about this: we know D4 has a normal subgroup, . what does this mean geometrically? well, imagine that we have a square made out of "really stretchy material". if we "mod out the rotations", what we are really doing is identifying all 4 sides,. so imagine "curving the square", blowing it up like a parachute, and then drawing the edges tight (like with a drawstring), so that our square has now become a sphere. our diagonals are now two circles on the sphere: one going from the north pole to the south pole, another at the equator. the coset is now is the sphere turning on its axis, and the coset s is the sphere turning "end over end" (flipping the poles).