Combinatorics Through Guided Discovery, 2004a

114 6. Groups acting on sets
⇒ ·
Problem 278. (Relevant to Appendix C.) A permutation σ is called an involution
if σ 2 = ι. When you write an involution as a product of disjoint
cycles, what is special about the cycles?
6.2 Groups Acting on Sets
We defined the rotation group R 4 and the dihedral group D 4 as groups of permutations
of the vertices of a square. These permutations represent rigid motions of
the square in the plane and in three dimensional space respectively. The square
has geometric features of interest other than its vertices; for example its diagonals,
or its edges. Any geometric motion of the square that returns it to its original
position takes each diagonal to a possibly different diagonal, and takes each edge
to a possibly different edge. In Figure 6.2.1 we show the results on the sides and
diagonals of the rotations of a square. The rotation group permutes the sides of
the square and permutes the diagonals of the square as it rotates the square. Thus,
we say the rotation group “acts” on the sides and diagonals of the square.
1 s 1 2 1 s 4 2
d 13
d 24
s 4 s 2 s 3 s 1
s 3 s 2
d 24
d 13
4
3 4
3
ρ
s 3
s 2
1 2 1 2
s 1 s 4 s 3
1 2
ρ 2 ρ 3 ρ 4 = identity
d 13
d 24
d 13
s 2 s 4 s 1 s 3 s 4 s 2
d 24
d 13
d 24
4
3 4
3 4
3
s 1
= ρ 0
Figure 6.2.1: The results on the sides and diagonals of rotating the square
Problem 279.
(a) Write down the two-line notation for the permutation ρ that a 90
degree rotation does to the sides of the square.
(b) Write down the two-line notation for the permutation ρ 2 that a 180
degree rotation does to the sides of the square.
(c) Is ρ 2 = ρ ◦ ρ? Why or why not?
(d) Write down the two-line notation for the permutation ̂ρ that a 90
degree rotation does to the diagonals d 13 and d 24 of the square.
(e) Write down the two-line notation for the permutation ̂ρ 2 that a 180
degree rotation does to the diagonals d 13 and d 24 of the square.
(f) Is ̂ρ 2 = ̂ρ ◦ ̂ρ? Why or why not? What familiar permutation is ̂ρ 2 in
this case?