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Group action - Groupprops

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The term group action or action of a group is used for the notion defined here. An alternative term sometimes used is permutation representation. A group action on the left (simply called a group action if the convention is of left actions) of a group on a set is a map such that the following two conditions are satisfied: A group action on a set or an action of a group on a set is a group homomorphism from the group to the symmetric group on the set. In symbols, a group action of a group on a set is a homomorphism where denotes the symmetric group on . Further information: Equivalence of definitions of group action For convenience, we omit the symbols or , and write the action of on as , or sometimes just as . We can then rewrite the first condition as: This is just like associativity, and hence we can drop the parenthesization, so we often write for either of the above. A group action on the right (simply called a group action if the convention is of right actions) of a gr

Group action - Groupprops Jump to content Want site search autocompletion? See here Encountering 429 Too Many Requests errors when browsing the site? See here From Groupprops This article is about a basic definition in group theory. The article text may, however, contain advanced material. VIEW : Definitions built on this | Facts about this: ( facts closely related to Group action , all facts related to Group action ) | Survey articles about this | Survey articles about definitions built on this VIEW RELATED : Analogues of this | Variations of this | Opposites of this | [SHOW MORE] View a comp

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