PERMUTATIONS 25

Permutations in General

We can play this game with any set A.Thegroup of permutations

of A is the set of functions from A to itself that are one-to-one

correspondences. Composition of permutations is deﬁned just as

above, as composition of functions.

The neutral permutation e in the group of permutations of A

is called the identity permutation. By deﬁnition, e(x) = x for every

x in A. For example, if A ={a, b, c},thene is the permutation

diagrammed by

a → a

(e): b → b

c → c.

To ﬁnd the inverse of any permutation, we switch the left and

right columns, leaving the arrows in place, and then, if we wish,

we reorder the rows so that the left-hand column is in the standard

order. In symbols, if f is a permutation of the set A,thenf

−1

is

the permutation deﬁned by f

−1

(x) = y if and only if f (y) = x.For

example, if g ◦ h is the permutation of {a, b, c} we computed in the

exercise on page 24, so that g ◦ h is given by the diagram

a → c

(g ◦ h): b → a

c → b,

then (g ◦ h)

−1

is given by the diagram

c → a

(g ◦ h)

−1

: a → b

b → c,

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