A basic question about cross product notation in group action
https://en.wikipedia.org/wiki/Group_action#cite_note-1
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
edited 9 mins ago
dmtri
1,3781521
asked 30 mins ago
Rodriguez
182
add a comment |
https://en.wikipedia.org/wiki/Group_action#cite_note-1
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
edited 9 mins ago
dmtri
1,3781521
asked 30 mins ago
Rodriguez
182
add a comment |
https://en.wikipedia.org/wiki/Group_action#cite_note-1
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
edited 9 mins ago
dmtri
1,3781521
asked 30 mins ago
Rodriguez
182
https://en.wikipedia.org/wiki/Group_action#cite_note-1
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
group-theory notation group-actions
edited 9 mins ago
dmtri
1,3781521
asked 30 mins ago
Rodriguez
182
edited 9 mins ago
dmtri
1,3781521
asked 30 mins ago
Rodriguez
182
edited 9 mins ago
dmtri
1,3781521
edited 9 mins ago
dmtri
1,3781521
edited 9 mins ago
dmtri
1,3781521
1,3781521
asked 30 mins ago
Rodriguez
182
asked 30 mins ago
Rodriguez
182
asked 30 mins ago
Rodriguez
182
182
add a comment |
add a comment |
3 Answers
3
active
oldest
votes
It is the Cartesian product of (the underlying set of) $G$ with $X$, i.e., $$Gtimes X={(g, x)mid gin G, xin X}.$$
answered 24 mins ago
Shaun
8,552113580
add a comment |
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).
answered 27 mins ago
Randall
9,07611129
add a comment |
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 $X$.
answered 26 mins ago
Alejandro Nasif Salum
4,209118
add a comment |
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3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
It is the Cartesian product of (the underlying set of) $G$ with $X$, i.e., $$Gtimes X={(g, x)mid gin G, xin X}.$$
answered 24 mins ago
Shaun
8,552113580
add a comment |
It is the Cartesian product of (the underlying set of) $G$ with $X$, i.e., $$Gtimes X={(g, x)mid gin G, xin X}.$$
answered 24 mins ago
Shaun
8,552113580
add a comment |
It is the Cartesian product of (the underlying set of) $G$ with $X$, i.e., $$Gtimes X={(g, x)mid gin G, xin X}.$$
answered 24 mins ago
Shaun
8,552113580
It is the Cartesian product of (the underlying set of) $G$ with $X$, i.e., $$Gtimes X={(g, x)mid gin G, xin X}.$$
answered 24 mins ago
Shaun
8,552113580
edited 13 mins ago
answered 24 mins ago
Shaun
8,552113580
answered 24 mins ago
Shaun
8,552113580
answered 24 mins ago
Shaun
8,552113580
8,552113580
add a comment |
add a comment |
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).
answered 27 mins ago
Randall
9,07611129
add a comment |
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).
answered 27 mins ago
Randall
9,07611129
add a comment |
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).
answered 27 mins ago
Randall
9,07611129
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).
answered 27 mins ago
Randall
9,07611129
answered 27 mins ago
Randall
9,07611129
answered 27 mins ago
Randall
9,07611129
answered 27 mins ago
Randall
9,07611129
9,07611129
add a comment |
add a comment |
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 $X$.
answered 26 mins ago
Alejandro Nasif Salum
4,209118
add a comment |
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 $X$.
answered 26 mins ago
Alejandro Nasif Salum
4,209118
add a comment |
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 $X$.
answered 26 mins ago
Alejandro Nasif Salum
4,209118
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 $X$.
answered 26 mins ago
Alejandro Nasif Salum
4,209118
answered 26 mins ago
Alejandro Nasif Salum
4,209118
answered 26 mins ago
Alejandro Nasif Salum
4,209118
answered 26 mins ago
Alejandro Nasif Salum
4,209118
4,209118
add a comment |
add a comment |
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