Combinatorics Through Guided Discovery, 2004a

6.2. Groups Acting on Sets 119
6.2.2 Orbits
• Problem 287. Refer back to Problem 282 in answering the following questions.
(a) What is the set of two element subsets that you get by computing
σ({1, 2}) for all σ in D 4 ?
(b) What is the multiset of two-element subsets that you get by computing
σ({1, 2}) for all σin D 4 ?
(c) What is the set of two-element subsets you get by computing σ({1, 3})
for all σ in D 4 ?
(d) What is the multiset of two-element subsets that you get by computing
σ({1, 3}) for all σ in D 4 ?
(e) Describe these two sets geometrically in terms of the square.
• Problem 288. This problem uses the notation for permutations in the dihedral
group of the square introduced before Problem 259. What is the effect
of a 180 degree rotation ρ 2 on the diagonals of a square? What is the effect
of the flip ϕ 1|3 on the diagonals of a square? How many elements of D 4
send each diagonal to itself? How many elements of D 4 interchange the
diagonals of a square?
In Problem 287 you saw that the action of the dihedral group D 4 on two element
subsets of {1, 2, 3, 4} seems to split them into two sets, one with two elements
and one with 4. We call these two sets the “orbits” of D 4 acting on the two
elements subsets of {1, 2, 3, 4}. More generally, the orbit of a permutation group G
determined by an element x of a set S on which G acts is
{σ(x)|σ ∈ G},
and is denoted by Gx. InProblem 287 it was possible to have Gx = Gy. In fact in
that problem, Gx = Gy for every y in Gx.
Problem 289. Suppose a group acts on a set S. Could an element of S be in
two different orbits? (Say why or why not.) (h)
Problem 289 almost completes the proof of the following theorem.
Theorem 6.2.3. Suppose a group acts on a set S. The orbits of G form a partition of S.
It is probably worth pointing out that this theorem tells us that the orbit Gx is
also the orbit Gy for any element y of Gx.