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I'm reading a group theory book and the author gives the following definition for cycle type:

If $\sigma \in S_n$ is the product of disjoint cycles of lengths $n_1,n_2,\ldots ,n_r$ with $n_1 \leq n_2 \leq \cdots \leq n_r$ (including its 1-cycles) then the sequence of integers $n_1, n_2, \ldots ,n_r$ is called the cycle type of $\sigma$ where $S_n$ is the symmetric group of degree n.

I'm curious about the significance of the cycle type. If two elements of $S_n$ have the same cycle type are they somehow related?

T. Fo
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  • Two elements with same cycle type are conjugates. So Their orders will be same as well. – Anurag A Jan 07 '19 at 19:42
  • Yes; two elements of $S_n$ are conjugate to one another if and only if they have the same cycle type. Also, you can determine whether an element of $S_n$ lies in $A_n$ depending on its cycle type, and you can use the cycle type to determine the order of the permutation. It's an important concept. – Arturo Magidin Jan 07 '19 at 19:42
  • I know that you did not ask for a proof, but the answers at the duplicate are so excellent, that it is worth reading them, also for this question. – Dietrich Burde Jan 07 '19 at 19:43
  • You will find a proof of what Arturo wrote in his comment here, written by… Arturo himself. :-) – José Carlos Santos Jan 07 '19 at 19:45

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