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Fearless Symmetry by Robert Gross, Avner Ash

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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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