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In mathematics, a symmetry group is an abstraction used to describe the symmetries of an object. A group action formalizes the relationship between the group and the symmetries of the object. It relates each element of the group to a particular transformation of the object. In this case, the group is also called a permutation group (especially if the set is finite or not a vector space) or transformation group (especially if the set is a vector space and the group acts like linear transformations of the set). A permutation representation of a group ''G'' is a representation of ''G'' as a group of permutations of the set (usually if the set is finite), and may be described as a group representation of ''G'' by permutation matrices. It is the same as a group action of ''G'' on an ''ordered'' basis of a vector space. A group action is an extension to the notion of a symmetry group in which every element of the group "acts" like a bijective transformation (or "symmetry") of some set, without being identified with that transformation. This allows for a more comprehensive description of the symmetries of an object, such as a polyhedron, by allowing the same group to act on several different sets of features, such as the set of vertices, the set of edges and the set of faces of the polyhedron. If ''G'' is a group and ''X'' is a set, then a group action may be defined as a group homomorphism ''h'' from ''G'' to the symmetric group on ''X''. The action assigns a permutation of ''X'' to each element of the group in such a way that the permutation of ''X'' assigned to * the identity element of ''G'' is the identity transformation of ''X''; * a product ''gk'' of two elements of ''G'' is the composition of the permutations assigned to ''g'' and ''k''. The abstraction provided by group actions is a powerful one, because it allows geometrical ideas to be applied to more abstract objects. Many objects in mathematics have natural group actions defined on them. In particular, groups can act on other groups, or even on themselves. Because of this generality, the theory of group actions contains wide-reaching theorems, such as the orbit stabilizer theorem, which can be used to prove deep results in several fields. == Definition == If ''G'' is a group and ''X'' is a set, then a (''left'') ''group action'' ''φ'' of ''G'' on ''X'' is a function : that satisfies the following two axioms (where we denote ''φ''(''g'', ''x'') as ''g''.''x''): ; Identity: for all ''x'' in ''X''. (Here, ''e'' denotes the neutral element of the group ''G''.) ; Compatibility: for all ''g'', ''h'' in ''G'' and all ''x'' in ''X''. (Here, ''gh'' denotes the result of applying the group operation of ''G'' to the elements ''g'' and ''h''.) The group ''G'' is said to act on ''X'' (on the left). The set ''X'' is called a (''left'') ''G-set''. From these two axioms, it follows that for every ''g'' in ''G'', the function which maps ''x'' in ''X'' to ''g''.''x'' is a bijective map from ''X'' to ''X'' (its inverse being the function which maps ''x'' to ''g''−1.''x''). Therefore, one may alternatively define a group action of ''G'' on ''X'' as a group homomorphism from ''G'' into the symmetric group Sym(''X'') of all bijections from ''X'' to ''X''.〔This is done e.g. by 〕 In complete analogy, one can define a ''right group action'' of ''G'' on ''X'' as an operation mapping to ''x''.''g'' and satisfying the two axioms: ; Identity: for all ''x'' in ''X''. ; Compatibility: for all ''g'', ''h'' in ''G'' and all ''x'' in ''X''; The difference between left and right actions is in the order in which a product like ''gh'' acts on ''x''. For a left action ''h'' acts first and is followed by ''g'', while for a right action ''g'' acts first and is followed by ''h''. Because of the formula , one can construct a left action from a right action by composing with the inverse operation of the group. Also, a right action of a group ''G'' on ''X'' is the same thing as a left action of its opposite group ''G''op on ''X''. It is thus sufficient to only consider left actions without any loss of generality. Alternate Definition An equivalent definition that is often used involves symmetry groups. Let ''G'' be a group, let ''X'' be a set and let ''SX'' be the symmetry group of ''X'', the set of all permutations of elements of ''X''. ''G'' acts on ''X'' if there is a group homomorphism from ''G'' to ''SX''. :. If such a homomorphism exists, then both the properties identity and compatibility as described above come directly from properties of group homomorphisms. ; Identity: ''θ''(''e'') = ''ι''. (''θ'' maps the identity of ''G'' to the identity permutation.) ; Compatibility: ''σgh'' = ''θ''(''gh'') = ''θ''(''g'')''θ''(''h'') = ''σgσh''. (''θ'' respects both group operations.) 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Group action」の詳細全文を読む スポンサード リンク
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