Conjugacy classes of a symmetric group are determined by cycle structure

Two permutations $𝜎 1, 𝜎 2 \in 𝑆 𝑛$ are conjugate iff they have the same number of $𝑗$ -cycles $𝑘 𝑗 (𝜎 1) = 𝑘 𝑗 (𝜎 2)$ for each $𝑗 = 1, \dots, 𝑛$ . group

Proof

Let $𝜎, 𝜏 \in 𝑆 𝑛$ where $𝜎$ the product of $𝑚$ disjoint cycles
$𝜎 = 𝛼 1 𝛼 2 \dots 𝛼 𝑚 - 1 𝛼 𝑚$
Then conjugating $𝜎$ by $𝜏$ is the same as the product of conjugating each cycle
$𝜏 𝜎 𝜏 - 1 = 𝜏 𝛼 1 𝜏 - 1 𝜏 𝛼 2 𝜏 - 1 \dots 𝜏 𝛼 𝑚 - 1 𝜏 - 1 𝜏 𝛼 𝑚 𝜏 - 1$
but The conjugate of an n-cycle is an n-cycle, hence the cycle structure of $𝜏 𝛼 𝜏 - 1$ is identical.

The conjugacy classes of $𝑆 𝑛$ thus correspond to partitions of $𝑛$ .

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Conjugacy classes of a symmetric group are determined by cycle structure

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