A basic question about cross product notation in group action












2














In wiki, a group action is defined as




If $G $ is a group and $X$ is a set, then a (left) group action $φ$ of $G$ on $X$
is a function $varphicolon G times X to Xcolon (g,x)mapsto varphi(g,x)$




My question is, what is the meaning of cross product "$times$" here? Is this a direct product? If so, it is defined via two groups, but $X$ here is a set than group










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    2














    In wiki, a group action is defined as




    If $G $ is a group and $X$ is a set, then a (left) group action $φ$ of $G$ on $X$
    is a function $varphicolon G times X to Xcolon (g,x)mapsto varphi(g,x)$




    My question is, what is the meaning of cross product "$times$" here? Is this a direct product? If so, it is defined via two groups, but $X$ here is a set than group










    share|cite|improve this question



























      2












      2








      2







      In wiki, a group action is defined as




      If $G $ is a group and $X$ is a set, then a (left) group action $φ$ of $G$ on $X$
      is a function $varphicolon G times X to Xcolon (g,x)mapsto varphi(g,x)$




      My question is, what is the meaning of cross product "$times$" here? Is this a direct product? If so, it is defined via two groups, but $X$ here is a set than group










      share|cite|improve this question















      In wiki, a group action is defined as




      If $G $ is a group and $X$ is a set, then a (left) group action $φ$ of $G$ on $X$
      is a function $varphicolon G times X to Xcolon (g,x)mapsto varphi(g,x)$




      My question is, what is the meaning of cross product "$times$" here? Is this a direct product? If so, it is defined via two groups, but $X$ here is a set than group







      group-theory notation group-actions






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited 36 mins ago









      Shaun

      8,582113580




      8,582113580










      asked 1 hour ago









      Rodriguez

      183




      183






















          3 Answers
          3






          active

          oldest

          votes


















          6














          It is the Cartesian product of (the underlying set of) $G$ with $X$, i.e., $$Gtimes X={(g, x)mid gin G, xin X}.$$






          share|cite|improve this answer































            3














            It is the Cartesian product of the two sets $G$ and $X$. The group structure on $G$ will then be used in the definition of an "action" (associativity and identity acts like an identity).






            share|cite|improve this answer





























              2














              The $times$ here is not a cross product, but a cartesian product (an operation between sets). Here it indicates that the action $varphi$ is actually a function that has the set $G times X$ for its domain ($G times X$ is the set of all pairs of an element of $G$ together with an element of $X$). Its codomain is $X$.



              Or put it more simple, $phi$ takes one element of the group $G$ and one element of the set $X$, and returns as output an element of $X$.






              share|cite|improve this answer





















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                3 Answers
                3






                active

                oldest

                votes








                3 Answers
                3






                active

                oldest

                votes









                active

                oldest

                votes






                active

                oldest

                votes









                6














                It is the Cartesian product of (the underlying set of) $G$ with $X$, i.e., $$Gtimes X={(g, x)mid gin G, xin X}.$$






                share|cite|improve this answer




























                  6














                  It is the Cartesian product of (the underlying set of) $G$ with $X$, i.e., $$Gtimes X={(g, x)mid gin G, xin X}.$$






                  share|cite|improve this answer


























                    6












                    6








                    6






                    It is the Cartesian product of (the underlying set of) $G$ with $X$, i.e., $$Gtimes X={(g, x)mid gin G, xin X}.$$






                    share|cite|improve this answer














                    It is the Cartesian product of (the underlying set of) $G$ with $X$, i.e., $$Gtimes X={(g, x)mid gin G, xin X}.$$







                    share|cite|improve this answer














                    share|cite|improve this answer



                    share|cite|improve this answer








                    edited 1 hour ago

























                    answered 1 hour ago









                    Shaun

                    8,582113580




                    8,582113580























                        3














                        It is the Cartesian product of the two sets $G$ and $X$. The group structure on $G$ will then be used in the definition of an "action" (associativity and identity acts like an identity).






                        share|cite|improve this answer


























                          3














                          It is the Cartesian product of the two sets $G$ and $X$. The group structure on $G$ will then be used in the definition of an "action" (associativity and identity acts like an identity).






                          share|cite|improve this answer
























                            3












                            3








                            3






                            It is the Cartesian product of the two sets $G$ and $X$. The group structure on $G$ will then be used in the definition of an "action" (associativity and identity acts like an identity).






                            share|cite|improve this answer












                            It is the Cartesian product of the two sets $G$ and $X$. The group structure on $G$ will then be used in the definition of an "action" (associativity and identity acts like an identity).







                            share|cite|improve this answer












                            share|cite|improve this answer



                            share|cite|improve this answer










                            answered 1 hour ago









                            Randall

                            9,07611129




                            9,07611129























                                2














                                The $times$ here is not a cross product, but a cartesian product (an operation between sets). Here it indicates that the action $varphi$ is actually a function that has the set $G times X$ for its domain ($G times X$ is the set of all pairs of an element of $G$ together with an element of $X$). Its codomain is $X$.



                                Or put it more simple, $phi$ takes one element of the group $G$ and one element of the set $X$, and returns as output an element of $X$.






                                share|cite|improve this answer


























                                  2














                                  The $times$ here is not a cross product, but a cartesian product (an operation between sets). Here it indicates that the action $varphi$ is actually a function that has the set $G times X$ for its domain ($G times X$ is the set of all pairs of an element of $G$ together with an element of $X$). Its codomain is $X$.



                                  Or put it more simple, $phi$ takes one element of the group $G$ and one element of the set $X$, and returns as output an element of $X$.






                                  share|cite|improve this answer
























                                    2












                                    2








                                    2






                                    The $times$ here is not a cross product, but a cartesian product (an operation between sets). Here it indicates that the action $varphi$ is actually a function that has the set $G times X$ for its domain ($G times X$ is the set of all pairs of an element of $G$ together with an element of $X$). Its codomain is $X$.



                                    Or put it more simple, $phi$ takes one element of the group $G$ and one element of the set $X$, and returns as output an element of $X$.






                                    share|cite|improve this answer












                                    The $times$ here is not a cross product, but a cartesian product (an operation between sets). Here it indicates that the action $varphi$ is actually a function that has the set $G times X$ for its domain ($G times X$ is the set of all pairs of an element of $G$ together with an element of $X$). Its codomain is $X$.



                                    Or put it more simple, $phi$ takes one element of the group $G$ and one element of the set $X$, and returns as output an element of $X$.







                                    share|cite|improve this answer












                                    share|cite|improve this answer



                                    share|cite|improve this answer










                                    answered 1 hour ago









                                    Alejandro Nasif Salum

                                    4,269118




                                    4,269118






























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