A question about “$times$” notation with group actions
Per Wikipedia, a (left) group action is defined as follows:
If $G $ is a group and $X$ is a set, then a (left) group action $φ$ of $G$ on $X$ is a function $$varphi : G times X to X : (g,x)mapsto varphi(g,x)cdots
$$ (and so on).
My question is, what is the meaning of the "$times$" notation in this context? It immediately calls to mind direct products and cross products. If so, it is defined via two groups, but $X$ here is a set than group, so I have my doubts.
group-theory notation group-actions
edited 2 days ago
Shaun
8,711113680
asked Dec 28 '18 at 4:28
Rodriguez
224
add a comment |
Per Wikipedia, a (left) group action is defined as follows:
If $G $ is a group and $X$ is a set, then a (left) group action $φ$ of $G$ on $X$ is a function $$varphi : G times X to X : (g,x)mapsto varphi(g,x)cdots
$$ (and so on).
My question is, what is the meaning of the "$times$" notation in this context? It immediately calls to mind direct products and cross products. If so, it is defined via two groups, but $X$ here is a set than group, so I have my doubts.
group-theory notation group-actions
edited 2 days ago
Shaun
8,711113680
asked Dec 28 '18 at 4:28
Rodriguez
224
1
For the record, this is not the definition of a left group action, the crucial part of the definition is missing.
– Did
Dec 28 '18 at 17:37
add a comment |
Per Wikipedia, a (left) group action is defined as follows:
If $G $ is a group and $X$ is a set, then a (left) group action $φ$ of $G$ on $X$ is a function $$varphi : G times X to X : (g,x)mapsto varphi(g,x)cdots
$$ (and so on).
My question is, what is the meaning of the "$times$" notation in this context? It immediately calls to mind direct products and cross products. If so, it is defined via two groups, but $X$ here is a set than group, so I have my doubts.
group-theory notation group-actions
edited 2 days ago
Shaun
8,711113680
asked Dec 28 '18 at 4:28
Rodriguez
224
Per Wikipedia, a (left) group action is defined as follows:
If $G $ is a group and $X$ is a set, then a (left) group action $φ$ of $G$ on $X$ is a function $$varphi : G times X to X : (g,x)mapsto varphi(g,x)cdots
$$ (and so on).
My question is, what is the meaning of the "$times$" notation in this context? It immediately calls to mind direct products and cross products. If so, it is defined via two groups, but $X$ here is a set than group, so I have my doubts.
group-theory notation group-actions
group-theory notation group-actions
edited 2 days ago
Shaun
8,711113680
asked Dec 28 '18 at 4:28
Rodriguez
224
edited 2 days ago
Shaun
8,711113680
asked Dec 28 '18 at 4:28
Rodriguez
224
edited 2 days ago
Shaun
8,711113680
edited 2 days ago
Shaun
8,711113680
edited 2 days ago
Shaun
8,711113680
8,711113680
asked Dec 28 '18 at 4:28
Rodriguez
224
asked Dec 28 '18 at 4:28
Rodriguez
224
asked Dec 28 '18 at 4:28
Rodriguez
224
224
1
For the record, this is not the definition of a left group action, the crucial part of the definition is missing.
– Did
Dec 28 '18 at 17:37
add a comment |
1
For the record, this is not the definition of a left group action, the crucial part of the definition is missing.
– Did
Dec 28 '18 at 17:37
1
1
For the record, this is not the definition of a left group action, the crucial part of the definition is missing.
– Did
Dec 28 '18 at 17:37
For the record, this is not the definition of a left group action, the crucial part of the definition is missing.
– Did
Dec 28 '18 at 17:37
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 Dec 28 '18 at 4:33
Shaun
8,711113680
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 Dec 28 '18 at 4:30
Randall
9,08111129
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 Dec 28 '18 at 4:31
Alejandro Nasif Salum
4,399118
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 Dec 28 '18 at 4:33
Shaun
8,711113680
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 Dec 28 '18 at 4:33
Shaun
8,711113680
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 Dec 28 '18 at 4:33
Shaun
8,711113680
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 Dec 28 '18 at 4:33
Shaun
8,711113680
edited Dec 28 '18 at 4:44
answered Dec 28 '18 at 4:33
Shaun
8,711113680
answered Dec 28 '18 at 4:33
Shaun
8,711113680
answered Dec 28 '18 at 4:33
Shaun
8,711113680
8,711113680
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 Dec 28 '18 at 4:30
Randall
9,08111129
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 Dec 28 '18 at 4:30
Randall
9,08111129
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 Dec 28 '18 at 4:30
Randall
9,08111129
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 Dec 28 '18 at 4:30
Randall
9,08111129
answered Dec 28 '18 at 4:30
Randall
9,08111129
answered Dec 28 '18 at 4:30
Randall
9,08111129
answered Dec 28 '18 at 4:30
Randall
9,08111129
9,08111129
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 Dec 28 '18 at 4:31
Alejandro Nasif Salum
4,399118
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 Dec 28 '18 at 4:31
Alejandro Nasif Salum
4,399118
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 Dec 28 '18 at 4:31
Alejandro Nasif Salum
4,399118
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 Dec 28 '18 at 4:31
Alejandro Nasif Salum
4,399118
answered Dec 28 '18 at 4:31
Alejandro Nasif Salum
4,399118
answered Dec 28 '18 at 4:31
Alejandro Nasif Salum
4,399118
answered Dec 28 '18 at 4:31
Alejandro Nasif Salum
4,399118
4,399118
add a comment |
add a comment |
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For the record, this is not the definition of a left group action, the crucial part of the definition is missing.
– Did
Dec 28 '18 at 17:37