39.3 Sign and Cycle Structure

The sign of a permutation perm is defined as (-1)^k where k is the number of cycles of perm of even length.

The sign is a homomorphism from the symmetric group onto the multiplicative group { +1, -1 }, the kernel of which is the alternating group.

is the cycle structure (i.e. the numbers of cycles of different lengths) of perm. This is encoded in a list l in the following form: The i-th entry of l contains the number of cycles of perm of length i+1. If perm contains no cycles of length i+1 it is not bound. Cycles of length 1 are ignored.

gap> SignPerm((1,2,3)(4,5));  
-1
gap> CycleStructurePerm((1,2,3)(4,5,9,7,8)); 
[ , 1,, 1 ]

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