Section 5.3
Alternating groups
From the theory in Section 5.2, every permutation can be written as a product of transpositions. To be able to distinguish between products of even and odd length, we need the following result.
Theorem
If a permutation is written in two ways as a product of transpositions, then both products have even length or both products have odd length.
If a permutation is written in two ways as a product of transpositions, then both products have even length or both products have odd length.
In other words, no permutation can be written both as a product
of transpositions of even length and as such a product of odd length.
So if one product involves an even (odd) number of factors, then all
products involve an even (odd) number of factors. This justifies
the following definition.
Definition
Let g
Sn be a permutation.
The sign (signum) of g,
denoted by sgn(g), is defined as
Let g
Sn be a permutation.
The sign (signum) of g,
denoted by sgn(g), is defined as
- 1, if g can be written as a product of an even number of 2-cycles, and
- -1, if g can be written as a product of an odd number of 2-cycles.
We say g is even if sgn(g) = 1
and odd if sgn(g) = -1.