MATH 433
March 6, 2015
Quiz 6: Solutions
1 2 3 4 5
1 2 3 4 5
Problem 1. Let π =
,σ=
. Find the cycle decomposition
3 2 5 1 4
2 1 4 5 3
for the permutations π, σ, and τ = πσ.
Solution: π = (1 3 5 4), σ = (1 2)(3 4 5), πσ = (1 2 3).
The two-row representation of the permutation πσ is obtained from the two-row representations of π
and σ as follows. We write π underneath σ, then reorder the columns of π so that the upper row of π
matches the lower row of σ, and then erase the matching rows:
1 2 3 4 5
σ=
2 1 4 5 3
1 2 3 4 5
=⇒ πσ =
2 3 1 4 5
2 1 4 5 3
π=
2 3 1 4 5
The cycle decomposition of a permutation is obtained from its two-row representation by inspection.
Problem 2. Find the order of the permutation π = (1 2)(2 3 5)(3 4 5 6).
Solution: π has order 4.
The permutation π is defined as a product of cycles that are not disjoint. To find the order of π, we
need to represent π as a product of disjoint cycles. Keeping in mind that the composition is evaluated from
the right to the left, we obtain π = (1 2 3 4)(5 6).
Since π is the product of two disjoint cycles of length 4 and 2, its order is the least common multiple of
4 and 2, which is 4.