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- Faithful_representation abstract "In mathematics, especially in the area of abstract algebra known as representation theory, a faithful representation ρ of a group G on a vector space V is a linear representation in which different elements g of G are represented by distinct linear mappings ρ(g). In more abstract language, this means that the group homomorphismρ: G → GL(V)is injective.Caveat: While representations of G over a field K are de facto the same as -modules (with denoting the group algebra of the group G), a faithful representation of G is not necessarily a faithful module for the group algebra. In fact each faithful -module is a faithful representation of G, but the converse does not hold. Consider for example the natural representation of the symmetric group Sn in n dimensions by permutation matrices, which is certainly faithful. Here the order of the group is n! while the n×n matrices form a vector space of dimension n2. As soon as n is at least 4, dimension counting means that some linear dependence must occur between permutation matrices (since 24 > 16); this relation means that the module for the group algebra is not faithful.".
- Faithful_representation wikiPageID "2617872".
- Faithful_representation wikiPageRevisionID "353697996".
- Faithful_representation hasPhotoCollection Faithful_representation.
- Faithful_representation id "F/f038170".
- Faithful_representation title "faithful representation".
- Faithful_representation subject Category:Representation_theory.
- Faithful_representation comment "In mathematics, especially in the area of abstract algebra known as representation theory, a faithful representation ρ of a group G on a vector space V is a linear representation in which different elements g of G are represented by distinct linear mappings ρ(g).".
- Faithful_representation label "Faithful representation".
- Faithful_representation sameAs m.07s2dt.
- Faithful_representation sameAs Q5431294.
- Faithful_representation sameAs Q5431294.
- Faithful_representation wasDerivedFrom Faithful_representation?oldid=353697996.
- Faithful_representation isPrimaryTopicOf Faithful_representation.