A basic question about cross product notation in group action
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 36 mins ago
Shaun
8,582113580
asked 1 hour ago
Rodriguez
183
add a comment |
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 36 mins ago
Shaun
8,582113580
asked 1 hour ago
Rodriguez
183
add a comment |
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 36 mins ago
Shaun
8,582113580
asked 1 hour ago
Rodriguez
183
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 36 mins ago
Shaun
8,582113580
asked 1 hour ago
Rodriguez
183
edited 36 mins ago
Shaun
8,582113580
asked 1 hour ago
Rodriguez
183
edited 36 mins ago
Shaun
8,582113580
edited 36 mins ago
Shaun
8,582113580
edited 36 mins ago
Shaun
8,582113580
8,582113580
asked 1 hour ago
Rodriguez
183
asked 1 hour ago
Rodriguez
183
asked 1 hour ago
Rodriguez
183
183
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 1 hour ago
Shaun
8,582113580
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 1 hour 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 1 hour ago
Alejandro Nasif Salum
4,269118
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 1 hour ago
Shaun
8,582113580
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 1 hour ago
Shaun
8,582113580
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 1 hour ago
Shaun
8,582113580
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 1 hour ago
Shaun
8,582113580
edited 1 hour ago
answered 1 hour ago
Shaun
8,582113580
answered 1 hour ago
Shaun
8,582113580
answered 1 hour ago
Shaun
8,582113580
8,582113580
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 1 hour 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 1 hour 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 1 hour 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 1 hour ago
Randall
9,07611129
answered 1 hour ago
Randall
9,07611129
answered 1 hour ago
Randall
9,07611129
answered 1 hour 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 1 hour ago
Alejandro Nasif Salum
4,269118
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 1 hour ago
Alejandro Nasif Salum
4,269118
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 1 hour ago
Alejandro Nasif Salum
4,269118
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 1 hour ago
Alejandro Nasif Salum
4,269118
answered 1 hour ago
Alejandro Nasif Salum
4,269118
answered 1 hour ago
Alejandro Nasif Salum
4,269118
answered 1 hour ago
Alejandro Nasif Salum
4,269118
4,269118
add a comment |
add a comment |
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