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## CHAPTER 7

### Permutation groups

#### 1 Cyclic decomposition

Let us recall that a cycle σ = (a1ar) in Sn is a permutation such that σ(ai) = ai + 1 for i = 1,…, r – 1, σ(ar) = a1, and σ(x) = x for every other x in n. Two cycles (a1,…,ar) (b1,bs) in Sn are disjoint permutations if and only if the sets {a1,…, ar} and {b1,…,bs} are disjoint. Note that a cycle of length r can be written in r ways, namely, as (a1 … ar) and (aiai+1ara1ai– 1), i = 2,…, r. A cycle of length r is also called an r-cycle.

1.1 Theorem. Any permutation is a product of pairwise disjoint cycles. This cyclic factorization is unique except for the order in which the cycles ...

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