Permutation representation

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In mathematics, the term permutation representation of a (typically finite) group can refer to either of two closely related notions: a representation of as a group of permutations, or as a group of permutation matrices. The term also refers to the combination of the two.

Abstract permutation representation[edit]

A permutation representation of a group on a set is a homomorphism from to the symmetric group of :

The image is a permutation group and the elements of are represented as permutations of .[1] A permutation representation is equivalent to an action of on the set :

See the article on group action for further details.

Linear permutation representation[edit]

If is a permutation group of degree , then the permutation representation of is the linear representation of

which maps to the corresponding permutation matrix (here is an arbitrary field).[2] That is, acts on by permuting the standard basis vectors.

This notion of a permutation representation can, of course, be composed with the previous one to represent an arbitrary abstract group as a group of permutation matrices. One first represents as a permutation group and then maps each permutation to the corresponding matrix.

References[edit]

  1. ^ Dixon, John D.; Mortimer, Brian (2012-12-06). Permutation Groups. Springer Science & Business Media. pp. 5–6. ISBN 9781461207313. 
  2. ^ Robinson, Derek J. S. (2012-12-06). A Course in the Theory of Groups. Springer Science & Business Media. ISBN 9781468401288.