Permutations

[gotmilk04]

1. The problem statement, all variables and given/known data
Let x=(1,2)(3,4) \in S_{8}.
Find an a \in S_{8} such that a-1xa=(5,6)(1,3)


2. Relevant equations



3. 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.