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(1) If s, tŒS n , then D s t = (D t ) s .
(2) For each transposition t in S n , D t =-D.
Using (1) and (2) we see that for the s in the theorem
s
= ()
DD
=
1
D
s
t
t
◊◊◊
t
12
s
t
= ()
=
D
1
D
.
hh
◊◊◊
h
12
t
In other words, (-1) s
= (-1) t and the theorem is proved.
Theorem B.3.2 shows that the next definition is well defined.
Definition. A permutation s in S n is said to be even if it can be written as a product
of an even number of transpositions. Otherwise, s is said to be odd .
Clearly, the product of two even permutations is even. Also, s in S n is even if and
only if s -1
t -1 is even, then ~ is an
equivalence relation on S n and S n gets partitioned into two equivalence classes by ~,
namely, the even and odd permutations.
is even. Therefore, if we define s~twhenever s
Definition.
The sign of a permutation s, denoted by sign (s), is defined by
() =+
=-
sign
s
1
1
,
if
s
s
is even
,
,
if
is odd
.
A function f : X d
Definition.
Æ Y is said to be symmetric if
(
) =
(
)
fx x
,
,...,
x
fx
x
,...,
x
()
()
( )
12
d
ss
1
,
2
s
d
for all x i ΠX and all permutations s of {1,2, ...,d}.
B.4
Groups
The word “group” comes up in several places in this topic. The concept is really only
essential in Chapters 7 and 8, which involve algebraic topology, but it is useful else-
where because it does capture some important properties in a single word. This
section will only survey those definitions and results that are needed in this topic.
More is not possible. In fact, with the exception of the fundamental group of a topo-
logical space which involves free groups and references the concept of the commuta-
tor subgroup, all the groups will be abelian. For that reason, other than giving the
necessary definitions, most of the discussion and examples will concentrate on abelian
groups. The interested reader is directed to topics on modern algebra for a more thor-
ough discussion of groups, in particular, the references in the bibliography.
Let G be a set and let · be a binary operation on G, that is, · is a map
:
GG G
¥
Æ
.
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