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Conjugacy classes in $ S_n$

Two permutations $ \alpha, \beta \in S_n$ are conjugate iff $ \exists \pi
\in S_n$ such that $ \alpha = \pi \beta \pi^{-1}$.

Theorem 5.6.10   Two permutations are conjugate iff they have the same cycle type.

This theorem is proved in the Algebra and Geometry course. We note the corollary that the number of conjugacy classes in $ S_n$ equals the number of partitions of $ n$.

Remark 5.6.11 (Determinants of an $ n \times n$ matrix)   In the Linear Maths course you will prove that if $ A = (a_{ij})$ is an $ n \times n$ matrix then

$\displaystyle \det A = \sum_{\pi \in S_n} \sign \pi \prod_{j=1}^n a_{j\, \pi(j)}.$    

root 2002-06-10