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I have heard that two permutations are conjugate if they have the same cyclic structure. Is there an intuitive way to understand why this is?

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    Have you read a proof of this fact, and found it to be unintuitive?2011-06-28
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    They're not: $(1 2) (1) = (1) (1 2) \implies (1 2)$ is conjugate with $(1)$ even though they have different cycle structure.2017-11-13
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    @Zaz That's not how conjugates are defined. If $a b = b c \implies b^{-1} a b = c \text{ for } a,b,c \in G$ then $a$ and $c$ are said to be conjugates. You're example just shows that $(12)$ and $(12)$ are conjugates i.e. they lie in the same conjugacy class, which is trivial. For $(12)$ to be conjugate with $(1)$ you have to show the existence of some $g \in G$ such that $g^{-1} (12) g = (1)$ and no such $g$ exists, hence $(12)$ and $(1)$ are not conjugates.2018-09-15

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