Math Help - Conjugacy

1. Conjugacy

hey I was having some trouble here and I was wondering if anybody could assist me.

Show that (1 2 3 4 5) is conjugate to (1 3 5 2 4) in $S_5$ but not in $A_5$.

Clearly (4 5 3 2) will work and this is odd, so not in A5. but why can't we find an even permutation that will work for the conjugation?

2. I found an old past paper solution that says the following:

If $^{\sigma}(1\ 2\ 3\ 4\ 5) = (1\ 3\ 5\ 2\ 4)$

Then $\sigma = (2\ 3\ 4\ 5)(1\ 2\ 3\ 4\ 5)^{l}$.

Why? And why is the number of elements in $A_5$ of order 5 equal to $\frac{5!}{5}$ ??

3. Originally Posted by slevvio
hey I was having some trouble here and I was wondering if anybody could assist me.

Show that (1 2 3 4 5) is conjugate to (1 3 5 2 4) in $S_5$ but not in $A_5$.

Clearly (4 5 3 2) will work and this is odd, so not in A5. but why can't we find an even permutation that will work for the conjugation?
Have you checked to see if the first is odd and the second even?

4. first and second what?

5. Originally Posted by slevvio
hey I was having some trouble here and I was wondering if anybody could assist me.

Show that (1 2 3 4 5) is conjugate to (1 3 5 2 4) in $S_5$ but not in $A_5$.

Clearly (4 5 3 2) will work and this is odd, so not in A5. but why can't we find an even permutation that will work for the conjugation?
Originally Posted by slevvio
first and second what?
Well, my first inclination (and this may be incorrect) was to use the fact that the product of any number of even permutations is even (and the odd analogue) to show that $g(12345)g^{-1}=(13524)\implies g\text{ is odd}$. You could do this by showing that $(12345)$ is even and $(13524)$ is odd. The problem would follow.

6. Those 2 permutations can be expressed as an even number of transpositions hence they are both even