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Permutation representation (disambiguation)
In mathematics, a permutation representation of a (typically finite) group $G$ can refer to two closely related notions: a representation of $G$ as a group of permutations, or as a group of permutation matrices.
==Abstract permutation representation==

A permutation representation of a group $G$ on a set $X$ is a homomorphism from $G$ to the symmetric group of $X$:
:$\rho\colon G \to \operatorname\left(X\right).$
The image $\rho\left(G\right)\sub \operatorname\left(X\right)$ is a permutation group and the elements of $G$ are represented as permutations of $X$. A permutation representation is equivalent to an action of $G$ on the set $X$:
:$G\times X \to X.$
See the article on group action for further details.

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