Advanced Abstract Algebra - Maharshi Dayanand University, Rohtak

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ADVANCED ABASTRACT ALGEBRA
f = (y 1 y 2 … y k ) o g
= (y 1 y 2 …. y k ) α 1 α 2 … α m
Hence, every permutation can be expressed as a composite of disjoint cycles.
For example, let
1
f =
6
2
4
3
7
4
2
5
5
6
1
7
8
8
9
9
3
be a permutation. Here 5 is a fixed element. Therefore, (5) is a cycle of length 1. Cycles of length 2 are
(1 6) and (2 4) whereas (3 7 8 9) is a cycle of length 4. Hence
f = (5) (1 6) (2 4) (3 7 8 9)
Theorem 16. Symmetric group S n is generated by transpositions, i.e., every permutation in S n is a
product of transpositions.
Proof. We have proved above that every permutation can be expressed as the composition of disjoint
cycles. Consider the m-cycle (x 1 , x 2 ,…, x m ). A simple computation shows that
(x 1 x 2 …. x m ) = (x 1 x m ) …. (x 1 x 3 ) (x 1 x 2 ),
that is, every cycle can be expressed as a product of transposition. Hence every permutation α ∈ S n can
be expressed as a product of transpositions.
Remark. The above decomposition of a cycle as the product of transposition is not unique. For
example,
(1 2 3) = (1 3) (1 2) = (3 2) (3 1)
However, it can be proved that the number of factors in the expression is always even or always odd.
Definition. A permutation is called even if it is a product of an even number of transpositions.
Similarly, a permutation is called odd if it is a product of odd number of transpositions.
Further,
(i) The product of two even permutations is even.
(ii) The product of two odd permutations is even.
(iii) The product of one odd and one even permutation is odd.
(iv) The inverse of an even permutation is an even permutation.
Theorem 17. If a permutation is expressed as a product of transpositions, then the number of
transpositions is either even in both cases or odd in both cases.
Proof. Let a permutation σ be expressed as the product of transpositions as given below:
This yields
σ = α 1 α 2 …. α r = β 1 β 2 … β s
−1
e = α 1 α 2 … α r β − 1 1
s β −
β
r − 1
1
= α 1 α 2 … α r β s β s−1 …. β 2 β 1 ,
since inverse of transposition is the transposition itself. The left side, that is, identity permutation is even
and therefore the right hand should also be an even permutation. Thus r+s is even which is possible if r
and s are both even or both odd. This completes the proof of the theorem.