Permutation Conjugation and Order

[fk378]

Homework Statement

Prove that there is no such permutation a such that
(a-inverse)*(1,2)*(a) = (3,4)(1,5)

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?

 

[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.

 

[fk378]

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