Advanced Abstract Algebra - Maharshi Dayanand University, Rohtak

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ADVANCED ABSTRACT ALGEBRA
Permutations:
Let S be a non-empty set/ A permutation of a set S is a function from S to S which is both one-to-one and
onto.
A permulation group of a set S is a set of permutations of S that forms a group under function composition.
Example 5:
Let
Define a permutation σ by
This 1–1 and onto mapping
can be written as
Define another permutation
Then
φ
( 1 ) = 3, φ( 3) = 2, φ( 2) = 1, φ( 4) = 4
σ
(φ
S
σφ
1
φσ =
3
2
1
3
2
4
4
1
2
2
3
3
4
4
1
1
=
1
2
2
3
4
4
3
The multiplication is from right to left.
We see ( φσ)( 1 ) = φ( σ( 1)
) = φ( 2) = 1,
and
.
Example 6: Symetric Groups
Let S3
denote the set of all one-to-one function from {1, 2, 3} to itself. Then S3
is a group of six elements,
under composition of mappings. These six elements are
1
e =
1
2
2
3
1
,
α =
3
2
2
3
3
,
α
1
2
1
=
3
2
1
3
2