Permutations (last question of sheet, yay )

[karnten07]

Permutations (last question of sheet, yay!)

1. Homework Statement [/b]

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

b.) Let n $$\geq$$1. Let <a1,...,as> $$\in$$Sn be a cycle and let $$\sigma$$$$\in$$Sn be arbitrary. Show that

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

Homework Equations

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 haven't because I am 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 $$\in$$Z and for odd numbers, that is n/2$$\notin$$Z to distinguish between the two cases.

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