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?
Why are two permutations conjugate iff they have the same cycle structure?
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abstract-algebra
group-theory
permutations
symmetric-groups
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8Have you read a proof of this fact, and found it to be unintuitive? – 2011-06-28
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0They'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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0@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