Advanced Abstract Algebra - Maharshi Dayanand University, Rohtak

UNIT-1 27
In this case the length of the cycle is 3. We can denote this permutation by (a b c).
Definition. A cyclic permutation of length 2 is called a Transposition.
1
For example,
1
2
3
3
is a transposition.
2
Definition. Two cycles are said to be disjoint if they have no object in common.
Definition. Two permutations α, β ∈ S n are called disjoint if
for all x ∈ S.
α(x) = x
β(x) ≠ x
α(x) ≠ x β(x) = x
In other words, α and β are disjoint if every x ∈ S moved by one permutation is fixed by the other.
Further, if α and β are disjoint permutations, then αβ = βα. For example, if we consider
then αβ = βα .
1
α =
2
2
3
3
1
2 3
, β =
1
1
2 3
Definition. A permutation α ∈ S n is said to be regular if either it is the identity permutation or it has
no fixed point and is the product of disjoint cycles of the same length.
For example,
1
2
2
3
is a regular permutation.
3
1
4
5
5
6
6
= (1 2 3) (4 5 6)
4
Theorem 15. Every permutation can be expressed as a product of pairwise disjoint cycles.
Proof. Let S = {x 1 , x 2 ,…, x n } be a finite set having n elements and f ∈ S n . If f is already a cycle, we are
through. So, let us suppose that f is not a cycle. We shall prove this theorem by induction on n.
If n = 1, the result is obvious. Let the theorem be true for a permutation of a set having less than n
elements. Then there exists a positive integer k < n and distinct elements y 1 ; y 2 ,…, y k in {x 1 , x 2 ,…, x n }
such that
f(y 1 ) = y 2
f(y 2 ) = y 3
………..
………..
f(y k−1 ) = y k
f(y k ) = y 1
Therefore (y 1 y 2 …. y k ) is a cycle of length k. Next, let g be the restriction of f to
T = {x 1 , x 2 ,…, x n } − {y 1 , y 2 ,…, y k }
Then g is a permutation of the set T containing n−k elements. Therefore, by induction hypothesis,
g = α 1 α 2 … α m ,
where α 1 , α 2 ,…, α m are pairwise disjoint cycles. But