I'm sitting with an exercise I can't quite finish.
Let be defined as follows: (of course it's not actually a fraction, I just dont know how to write a permutation.)
I have already answered the first three exercises:
(a) - Write as a product of disjoint cycles : .
(b) - Determine and the order of : and so the order of is 4.
(c) - How many simple transpositions are needed to write as a product of simple transpositions : From (b) I had that and that is the number of simple transitions needed.
The exercise I can't quite figure out is this:
(d) - Find , the alternating group, so .
Could someone give me a tip on how to solve this? Thanks on beforehand.
You did not define the sign of a permutation. Presumably, you are using the definition that is if can be written as a product of k transpositions. For my money, this is an awkward definition of the sign; however, it is a workable definition. (The problem comes with showing sgn is well defined.) An important fact about sgn is that
Now any cycle of length n can be written as a product of n-1 transpositions; e.g. (2 5 6 3) = (2 3)(2 6)(2 5) (composition is right to left), so sgn(2 5 6 3) = (-1)3 = -1. So the sign of your is 1; i.e. . I don't know what 8 has to do with this.
You correctly found the order of .
Now if is any cycle and , the conjugate is the cycle where for each letter i in the cycle, i is replaced by . Example: so Then . I leave it to you to verify this fact.
You should be able to use the above paragraph to find a with . Just write down and write a in two rowed form; figure out what the second row of should be.
First of all thanks. You made me realize how to solve the exercise!
Second, about the sign and my solution. We have a nifty algorithm for finding . Something about inversion count. This can then be used to find the sign of by calculating .
Anyway, thanks a million for the help!
Ah, you are using the "correct" definition of sign. Namely, if is a permutation, where is defined by
That is, is the number of pairs (i,j) with i < j such that or the number of inversions of . With this definition, it's "easy" to prove that if the permutation is a product of k transpositions, then the sgn(k)=(-1)k.