How Do You Find the Permutation 'a' in S8 for a Given Conjugation?

[gotmilk04]

Homework Statement

Let x=(1,2)(3,4) $$\in S_{8}$$.
Find an a $$\in S_{8}$$ such that a^-1xa=(5,6)(1,3)

Homework Equations

The Attempt at a Solution

I have no idea how you go about finding the a. Help please.

 

[jbunniii]

First, notice that $x$ does not have any effect on 5,6,7,8. Therefore, whichever inputs are mapped by $a$ to these numbers will be mapped back where they started by $a^{-1}$. You want $a^{-1}xa$ to leave 2,4,7,8 where they are, so you could for example define

$$a(2) = 5$$
$$a(4) = 6$$
$$a(7) = 7$$
$$a(8) = 8$$

Now let's look at the remaining numbers. Suppose we arbitrarily choose

$$a(1) = 1$$

Then $x$ maps 1 to 2, so $xa$ maps 1 to 2. We want $a^{-1}xa$ to map 1 to 3, therefore $a^{-1}$ must map 2 to 3:

$$a^{-1}(2) = 3$$

and thus

$$a(3) = 2$$

Thus far we have defined $a$ for six of the inputs, and it's easy to verify that $a^{-1}xa$ sends these six inputs to the right outputs. So now you have to define $a$ for the remaining two inputs (5 and 6). I'll let you take it from here.

Note that there are many possible solutions to this problem.