permutations

Question 1

What is a permutation

Answer 1

Let X be any set. Then a permutation on X is a bijection π: X → X.

Question 2

how do you find the inverse of a permutation written in cycle notation?

Answer 2

To find the inverse of permutation writ- ten in cycle notation, simply reverse the elements in each cycle (and rewrite to start with the smallest in each cycle

Question 3

For any permutation π ∈ Sn, there is some m > 0 such that πm = id and the permutations id, π, π2, . . . , πm−1 are all different.

Answer 3

For any permutation π ∈ Sn, there is some m > 0 such that πm = id and the permutations id, π, π2, . . . , πm−1 are all different.

Question 4

For a permutation π in Sn, the order of π is the smallest m > 0 such that πm = id.

Answer 4

For a permutation π in Sn, the order of π is the smallest m > 0 such that πm = id.

Question 5

Lemma 2.40 If π = σ1 …σt, where σ1,…,σt are disjoint cycles of lengths m1, . . . mt respectively, then the order of π is the least common multiple of these cycle lengths

Answer 5

Lemma 2.40 If π = σ1 …σt, where σ1,…,σt are disjoint cycles of lengths m1, . . . mt respectively, then the order of π is the least common multiple of these cycle lengths

Question 6

A permutation cannot be expressed as both a prod- uct of an even number of transpositions and of an odd number of transpositions.

Answer 6

A permutation cannot be expressed as both a prod- uct of an even number of transpositions and of an odd number of transpositions.