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# Def/Order (permutation)

Definition of Order of a permutation.: Suppose that $X$ is a finite set, and $\sigma$ is a permutation of $X$. Then, one may compose $\sigma$ with itself repeatedly, yielding permutations $\sigma^n = \sigma \circ \cdots \circ \sigma$, for any positive natural number $n$.

The order of $\sigma$ is the smallest positive integer $n$, such that $\sigma^n = Id_X$ (the identity permutation).

## Logical Connections

This definition logically relies on the following definitions and statements: Def/Permutation, Def/Identity function, State/Lagranges theorem, State/Every nonempty subset of N has a smallest element

The following statements and definitions logically rely on the material of this page: Def/Cyclic permutation

To visualize the logical connections between this definition and other items of mathematical knowledge, you can visit any of the following clusters, and click the "Visualize" tab: Clust/Permutations