Do these permutations commute?

Question

I read in my textbook that disjoint cycles commute. But what about $(1,2,3) = (1,3)(1,2)$ is it equal to $(1,2)(1,3) = (1,3,2)$ although they are not disjoint? And, if they are not equal how come in my textbook it says that $\langle(1,2,3)\rangle = \{(1),(1,2,3),(1,3,2) \}$ ?

Answer 1

The specific permutations are not equal. As the image of 1 is different, namely 2 and 3, respectively.

More generally, if you have two cycles that both start with 1, then they only give the same permutation if the string of numbers describing them is the same.

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The notation $\langle (1,2,3) \rangle$ means the subgroup generated by the permutation $(1,2,3)$. It is true that $(1,3,2)$ is an element of this. Indeed, $(1,3,2)$ and $(1,2,3)$ generate the same subgroup. Howvever, this does not mean the two elements are equal.

Answer 2

No, $\;(1\;2\;3)\neq (1\;3\;2)\;$ . In fact, the former one maps $\;1\;$ into $\;2\;$, whereas the latter maps it to $\;3\;$ .