Permutations

gotmilk04

1. The problem statement, all variables and given/known data
Let x=(1,2)(3,4) [tex]\in S_{8}[/tex].
Find an a [tex]\in S_{8}[/tex] 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 [itex]x[/itex] does not have any effect on 5,6,7,8. Therefore, whichever inputs are mapped by [itex]a[/itex] to these numbers will be mapped back where they started by [itex]a^{-1}[/itex]. You want [itex]a^{-1}xa[/itex] to leave 2,4,7,8 where they are, so you could for example define

[tex]a(2) = 5[/tex]
[tex]a(4) = 6[/tex]
[tex]a(7) = 7[/tex]
[tex]a(8) = 8[/tex]

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

[tex]a(1) = 1[/tex]

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

[tex]a^{-1}(2) = 3[/tex]

and thus

[tex]a(3) = 2[/tex]

Thus far we have defined [itex]a[/itex] for six of the inputs, and it's easy to verify that [itex]a^{-1}xa[/itex] sends these six inputs to the right outputs. So now you have to define [itex]a[/itex] 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.