UA-AUG-AUMAT229 Pre-read

We will consider a permutation as a function, so that the correspondence is unidirectional instead of bidirectional. (But we can always recover the bidirectional nature of a bijective correspondence using the inverse function.)

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Warning 8.1.3.

A permutation is (usually) not an isomorphism, as

the collection being permuted is not necessarily a group, and
even if the collection being permuted is a group, there is no requirement that the permutation be Operation-preserving map.

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As described at the beginning of Chapter 6 of the textbook, given a collection

X

we write

S_{X}

to denote the group of all permutations of

X,

where the group operation is function composition as usual. In the case that the collection being permuted is the

{1, 2, 3, \dots, n}

we write

S_{n}

instead.

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In any case, when

X

is a finite collection containing

n

objects, we can obtain an isomorphism between

S_{X}

and

S_{n}

by choosing a labelling of the objects in

X

by the numbers

1, 2, \dots, n .

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Example 8.1.4. Converting a permutation of letters into a permutation of numbers.

Consider $X = {a, b, c, d, e} .$ Let us relabel the elements of $X$ as

\begin{aligned} x_{1} & = a, & x_{2} & = b, & x_{3} & = c, & x_{4} & = d, & x_{5} & = e . \end{aligned}

At the beginning of this Pre-read section, we consider the permutation

(a c e) (b d)

of $X .$ With our relabelling, we can write this element of the group $S_{X}$ as

(x_{1} x_{3} x_{5}) (x_{2} x_{4}),

which is clearly essentially the same as the permutation

(1 3 5) (2 4)

in the group $S_{5} .$

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Remark 8.1.5.

What is really going on in Example 8.1.4 is that there are two bijective correspondences involved. First, the relabelling sets up a bijective correspondence between the collection of letters $X$ and the collection of numbers ${1, 2, 3, 4, 5} .$ Then the isomorphism (a special kind of bijective correspondence) between $S_{X}$ and $S_{5}$ is obtained by using the relabelling bijective correspondence to reinterpret a permutation of letters (i.e. a group element of $S_{X}$ ) as a permutation of numbers (i.e. a group element of $S_{5}$ ).

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Warning 8.1.6.

This isomorphic relationship $S_{X} \leftrightarrow S_{n}$ doesn't work if $X$ is not finite.

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Example 8.1.7. A permutation of an infinite collection.

The function $f : Z \to Z$ defined by $f (m) = - m$ is a bijective correspondence of $Z$ with itself that matches every integer with its negative. (And so $0$ is matched with itself.) But there is no way to interpret this permutation of $Z$ as a product of disjoint cycles in some $S_{n}$ permutation group, no matter how large $n$ is taken to be.