conjugacy in symmetric group

Question

I was reading Dummit and Foote and encountered the following statement: any two elements in $S_n$ are conjugate if and only if they have the same cycle types.

However, I am able to produce a counter example:

Let $(1 2 3)$ and $(4 5 6) (7 8)$ be in $S_{10}$, then $(4 5 6) (7 8) =(1 3 2) (4 5 6) (7 8) (1 2 3)$, which show that these two are conjugate.

What am I misunderstanding here?

Answer 1

Of course a permutation is conjugate to itself. It has the same cycle type as itself, as well. You could just as well have conjugated by the identity.

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Conjugation takes $k$-cycles to $k$-cycles: $\pi^{-1}(a_1\dots a_k)\pi=(\pi(a_1)\dots\pi(a_k))$.

Also, conjugation is a homomorphism. So, under conjugation, a product of cycles is the product of the conjugates. To finish, use that any permutation has a representation as a product of disjoint cycles.

Thus there can be no counterexample.

Answer 2

You have conjugated $(456)(78)$ by $(123)$, not shown that they are conjugate with each other.

For example, the conjugate of $(456)(78)$ by $(45)$ is

$$(45)(456)(78)(45)^{-1}=(546)(78),$$

meaning that $(456)(78)$ and $(546)(78)$ are conjugate with each other.

Answer 3

I think that you believe you found a counter example because you think that

$(456)(78)$ and $(132)(456)(78)(123)$ are two different cycle types of the same element.

This is not true since $(132)(456)(78)(123)$ is not a product of disjoint cycles.

(Maybe I have misunderstood your misunderstanding)