#Type/Definition #Topic/Linear_Algebra #Type/Theorem

Types: N/A
Examples: N/A
Constructions: 16.5 Permutation Definition of Determinant
Generalizations: 16.1 Permutation, 16.2 Symmetric Groups

Properties: N/A
Sufficiencies: N/A
Questions: N/A

Cycles

Let x1,x2,…,xk∈[n] such that xi≠xj for i,j∈[k] with k≤n and i≠j. A k-cycle is a 16.1 Permutation f of the form

π(x1)=x2,π(x2)=x3,…,π(xk−1=xk),π(xk)=x1

where π(i)=i for any i∈Sn−{x1,x2,…,xk}.

Existence of a Cycle Decomposition of Permutations

Let π:[n]→[n] be a permutation. Then, π can be written using a unique cycle decomposition.