22
1. ALGEBRAIC THEMES
Continue in this way until
has been written as a product of disjoint cy
cles.
Let me show you how to express the idea of the algorithm a little more
formally and also more concisely, using mathematical induction.
8
In the
preceding explanation, the phrase “continue in this way” is a signal that to
formalize the argument it is necessary to use induction.
Because disjoint cycles
1
and
2
commute (
1
2
D
2
1
, Exercise
1.5.13
), uniqueness in the following statement means uniqueness up to or
der; the factors are unique, and the order in which the factors are written
is irrelevant. Also note that
.a
1
; a
2
; : : : ; a
k
/
is the same cyclic permuta
tion as
.a
2
; : : : ; a
k
; a
1
/
, and there is no preferred first entry in the cycle
notation. Finally, in order not to have to make an exception for the identity
element
e
, we regard
e
as the product of the empty collection of cycles.
Theorem 1.5.3.
Every permutation of a finite set can be written uniquely
as a product of disjoint cycles.
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 Fall '08
 EVERAGE
 Logic, Algebra, Mathematical Induction, Natural number, disjoint cycles

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