Permutations (last question of sheet, yay!!)

[karnten07]

1. The problem statement, all variables and given/known data[/b]

\eta:=
(1 2 ... n-1 n)
(n n-1 ... 2 1)
\inS_{n} for any n\inN
n.b That should be 2 lines all in one large bracket btw
a.) Determine its sign.

b.) Let n \geq1. Let <a1,...,as> \inSn be a cycle and let \sigma\inSn be arbitrary. Show that

\sigma\circ <a1,...,as> \circ\sigma^{-1} = <\sigma(a1),...,\sigma(as)> in Sn.

2. Relevant equations



3. The attempt at a solution

I get the sign of the permutation to be (-1)^n/2

I don;t know how to do the second part, any ideas?

[karnten07]

Actually, i thought i had done the first part, but i havent because im stuck on how to show that for negative numbers, i want the n/2 to be taken as the rounded down value. For example if n=7 i want n/2 to be taken as 3. Is there is a simple way to do this for odd numbers but also keep the same form for positive values of n.

Also it should be (-1)^(n-2/2)

So it should be (-1)^(n-2/2) for even numbers of n and (-1)^(n-3/2) for odd values of n, is there a neater way to do this?

[karnten07]

I have just said for even numbers of n, that is, n/2 \inZ and for odd numbers, that is n/2\notinZ to distinguish between the two cases.

Im still thinking through part b.) so any help is welcomed.