An odd permutation of the corners implies an odd permutation of the centres and vice versa; however, even and odd permutations of the centres are indistinguishable due to the identical appearance of the pieces.
22.
An odd permutation of the corners implies an odd permutation of the centres and vice versa; however, even and odd permutations of the centres are indistinguishable due to the identical appearance of the pieces.
23.
This is equal to either + 1 or-1 depending on whether IJKL is either an even or odd permutation of 0123, respectively, and zero if any two indices take the same value.
24.
The odd permutations cannot form a subgroup, since the composite of two odd permutations is even, but they form a coset of " A " ( in " S " ).
25.
The odd permutations cannot form a subgroup, since the composite of two odd permutations is even, but they form a coset of " A " ( in " S " ).
26.
Any permutation of the edges in each set is possible, including odd permutations, giving 24 ! arrangements for each set or 24 ! 2 total, regardless of the position or orientation any other pieces.
27.
Any permutation of the intermediate and outer edges is possible, including odd permutations, giving 24 ! arrangements for each set or 24 ! 2 total, regardless of the position or orientation any other pieces.
28.
Though it may not be immediately obvious, the figure obtained by taking the even permutations with an even number of plus signs is the same as that obtained by taking the odd permutations with an odd number of plus signs.
29.
By defining the parity of \ sigma as the parity of N ( \ sigma ), a permutation that has an even length decomposition is an even permutation and a permutation that has one odd length decomposition is an odd permutation.
30.
There is just one exception to this : the alternating group " A " has outer automorphism group of order 4, rather than 2 as do the other simple alternating groups ( given by conjugation by an odd permutation ).
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