Combinatorics Through Guided Discovery, 2004a

112 6. Groups acting on sets
6.1.7 The cycle structure of a permutation
There is an even more efficient way to write down permutations. Notice that the
( )
1 2 3 4
product in Figure 6.1.2 is
. We have drawn the directed graph of this
2 3 1 4
permutation in Figure 6.1.5.
1
3
2
Figure 6.1.5: The directed graph of a permutation .
4
You see that the graph has two directed cycles, the rather trivial one with vertex 4
pointing to itself, and the nontrivial one with vertex 1 pointing to vertex 2 pointing
to vertex 3 which points back to vertex 1. A permutation is called a cycle if its
( )
( )
1 2 3
1 2 3 4
digraph consists of exactly one cycle. Thus
is a cycle but
2 3 1
2 3 1 4
is not a cycle by our definition. We write (1 2 3) or (2 3 1) or (3 1 2) to stand for the
( ) 1 2 3
cycle σ = .
2 3 1
We can describe cycles in another way as well. A cycle of the permutation σ
is a list (i σ(i) σ 2 (i) ... σ n (i)) that does not have repeated elements while the list
(i σ(i) σ 2 (i) ... σ n (i)) σ n+1 (i)) does have repeated elements.
Problem 272. If the list (i σ(i) σ 2 (i) ... σ n (i)) does not have repeated
elements but the list (i σ(i) σ 2 (i) ... σ n (i) σ n+1 (i)) does have repeated
elements, then what is σ n+1 (i)? (h)
We say σ j (i) is an element of the cycle (i σ(i) σ 2 (i) ... σ n (i)). Notice that the case
j =0means i is an element of the cycle. Notice also that if j > n, σ j (i) =σ j−n−1 (i),
so the distinct elements of the cycle are i, σ(i), σ 2 (i), through σ n (i). We think of
the cycle (i σ(i) σ 2 (i) ... σ n (i)) as representing the permutation σ restricted to
the set of elements of the cycle. We say that the cycles (i σ(i) σ 2 (i) ... σ n (i)) and
(j σ(j) σ 2 (j) ... σ n (j)) are equivalent if there is an integer k such that j = σ k (i).
• Problem 273. Find the cycles of the permutations ρ, ϕ 1|3 and ϕ 12|34 in the
group D 4 .