Note that the Vandermonde determinant is " alternating " in the entries, meaning that permuting the \ alpha _ i by an odd permutation changes the sign, while permuting them by an even permutation does not change the value of the determinant.
32.
S _ n can be embedded into A _ { n + 2 } by appending the transposition ( n + 1, n + 2 ) to all odd permutations, while embedding into A _ { n + 1 } is impossible for n > 1.
33.
Eleven can be flipped and arranged independently, giving 12 ! / 2 ?2 11 or 12 ! ?2 10 possibilities ( an odd permutation of the corners implies an odd permutation of the central edges, and vice versa, thus the division by 2 ).
34.
Eleven can be flipped and arranged independently, giving 12 ! / 2 ?2 11 or 12 ! ?2 10 possibilities ( an odd permutation of the corners implies an odd permutation of the central edges, and vice versa, thus the division by 2 ).
35.
Because this method generates permutations that alternate between being even and odd, it may easily be modified to generate only the even permutations or only the odd permutations : to generate the next permutation of the same parity from a given permutation, simply apply the same procedure twice.
36.
However, the determinant of a permutation matrix is the sign of the permutation, so to represent an odd permutation in "'SL " "'n ", we can take one of the nonzero elements to be " 1 instead of 1.
37.
For, except for, the automorphism group of A " n " is the symmetric group S " n ", with inner automorphism group A " n " and outer automorphism group Z 2; the outer automorphism comes from conjugation by an odd permutation.
38.
:: : If you've got the even ones, you can get the odd ones by picking any single odd permutation ( for example swapping the first and second element ) and applying it once to each even permutation in the list . talk ) 01 : 11, 25 July 2010 ( UTC)
39.
Considering } } as a subgroup of the symmetric group, } }, conjugation by any odd permutation is an outer automorphism of } } or more precisely " represents the class of the ( non-trivial ) outer automorphism of } } ", but the outer automorphism does not correspond to conjugation by any " particular " odd element, and all conjugations by odd elements are equivalent up to conjugation by an even element.
