Permutation conjugation

1. Oct 27, 2008

fk378

1. The problem statement, all variables and given/known data
Prove that there is no such permutation a such that
(a-inverse)*(1,2)*(a) = (3,4)(1,5)

3. The attempt at a solution
Does it have something to do with the order of (1,2)? I know the order is 2, so if we square (a-inverse)*(1,2)*(a), then we get the identity....how else can I think about it?

2. Oct 30, 2008

morphism

Do you know anything about cycle types? In particular, can you prove that they are invariant under conjugation?

If not, notice that 1->5 on the RHS. Convince yourself that (1,2) must send a(1) to a(5). Do the same for 3.

3. Oct 30, 2008

fk378

By a(1) and a(5) do you mean the numbers 1 and 5 in whatever the permutation a is?