## Permutations

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.
 Blog Entries: 1 Recognitions: Gold Member Homework Help Science Advisor 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.

 Tags permutation, permutations, symmetric group