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[/align]so we need to show that

.We first rewrite some of the factors in this product:

Note that this is equivalent to multiplying by

which means thatit leaves the value of the product unchanged. Doing this for alli,j, we can ensure that all the factors in the denominator are of the form for , with the corresponding factors in the numerator the same way round. Henceas required.

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[/align]so we need to show that

.We first rewrite some of the factors in this product:

Note that this is equivalent to multiplying by

which means thatas required.

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[/align]Lets look at the expression on the right-hand side. The denominator is a product of

factors, and so is the numerator. Let be a factor in the denominator. Either or . In the former case is a factor in the numberator, while in the latter case is a factor in the numerator. Conversely, if is a factor in the numerator, then (as ) either is a factor in the denominator (if ) or is a factor in the denominator (if ).So we see that if

is a factor in the denominator then either or is a factor in the numerator and conversely if is a factor in the numerator then either or is a factor in the denominator.Humphreyss definition, in terms of properties of an *n*-variable polynomial (which I incidentally recognize as the determinant of the Vandermonde matrix ), while not so intuitive, is more amenable to mathematical manipulation. Under this more flexible definition, the set of even permutations of *S[sub]n[/sub]* arises naturally as the kernel of a homomorphism from *S[sub]n[/sub]* to a 2-element group. This is thus a subgroup of index 2, called the alternating group *A[sub]n[/sub]* and it follows without more ado that the product of two even permutations is even.

Hence the identity is always an even permutation.

Also

A transposition

(where ) is a permutation which maps to and vice versa, leaving all other numbers fixed. Consider the ordered pairsUnder the transposition

, the image of the first component of each ordered pair is , which is greater than the second component of each ordered pair (which is fixed in the transposition)Hence

.Similarly, the ordered pairs

are in

, since the transposition maps the second component to while fixing the first component.The only other ordered pair that can belong to

is ., which is odd. ]]>A permutation is a bijection from the set to itself. The set of all permutations forms a group of order under composition of mappings, called the symmetric group of degree and denoted .

Permutations may be odd or even. Here is the most intuitive definition I can come up with for odd and even permutations. It is not the definition given in John F. Humphreyss book *A Course in Group Theory*, but it will do.

Given

, let

Then we say that

is even if the number of elements in is even, and odd if is odd. ]]>