A question about “$times$” notation with group actions
$begingroup$
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 Dec 29 '18 at 7:30
Shaun
9,241113684
asked Dec 28 '18 at 4:28
RodriguezRodriguez
224
$endgroup$
add a comment |
$begingroup$
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 Dec 29 '18 at 7:30
Shaun
9,241113684
asked Dec 28 '18 at 4:28
RodriguezRodriguez
224
$endgroup$
1
$begingroup$
For the record, this is not the definition of a left group action, the crucial part of the definition is missing.
$endgroup$
– Did
Dec 28 '18 at 17:37
add a comment |
$begingroup$
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 Dec 29 '18 at 7:30
Shaun
9,241113684
asked Dec 28 '18 at 4:28
RodriguezRodriguez
224
$endgroup$
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 Dec 29 '18 at 7:30
Shaun
9,241113684
asked Dec 28 '18 at 4:28
RodriguezRodriguez
224
edited Dec 29 '18 at 7:30
Shaun
9,241113684
asked Dec 28 '18 at 4:28
RodriguezRodriguez
224
edited Dec 29 '18 at 7:30
Shaun
9,241113684
edited Dec 29 '18 at 7:30
Shaun
9,241113684
edited Dec 29 '18 at 7:30
Shaun
9,241113684
9,241113684
asked Dec 28 '18 at 4:28
RodriguezRodriguez
224
asked Dec 28 '18 at 4:28
RodriguezRodriguez
224
asked Dec 28 '18 at 4:28
RodriguezRodriguez
224
224
1
$begingroup$
For the record, this is not the definition of a left group action, the crucial part of the definition is missing.
$endgroup$
– Did
Dec 28 '18 at 17:37
add a comment |
1
$begingroup$
For the record, this is not the definition of a left group action, the crucial part of the definition is missing.
$endgroup$
– Did
Dec 28 '18 at 17:37
1
1
$begingroup$
For the record, this is not the definition of a left group action, the crucial part of the definition is missing.
$endgroup$
– Did
Dec 28 '18 at 17:37
$begingroup$
For the record, this is not the definition of a left group action, the crucial part of the definition is missing.
$endgroup$
– Did
Dec 28 '18 at 17:37
add a comment |
3 Answers
3
active
oldest
votes
$begingroup$
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
ShaunShaun
9,241113684
$endgroup$
add a comment |
$begingroup$
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
RandallRandall
9,94111230
$endgroup$
add a comment |
$begingroup$
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 SalumAlejandro Nasif Salum
4,765118
$endgroup$
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
$begingroup$
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
ShaunShaun
9,241113684
$endgroup$
add a comment |
$begingroup$
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
ShaunShaun
9,241113684
$endgroup$
add a comment |
$begingroup$
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
ShaunShaun
9,241113684
$endgroup$
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
ShaunShaun
9,241113684
edited Dec 28 '18 at 4:44
answered Dec 28 '18 at 4:33
ShaunShaun
9,241113684
answered Dec 28 '18 at 4:33
ShaunShaun
9,241113684
answered Dec 28 '18 at 4:33
ShaunShaun
9,241113684
9,241113684
add a comment |
add a comment |
$begingroup$
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
RandallRandall
9,94111230
$endgroup$
add a comment |
$begingroup$
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
RandallRandall
9,94111230
$endgroup$
add a comment |
$begingroup$
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
RandallRandall
9,94111230
$endgroup$
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
RandallRandall
9,94111230
answered Dec 28 '18 at 4:30
RandallRandall
9,94111230
answered Dec 28 '18 at 4:30
RandallRandall
9,94111230
answered Dec 28 '18 at 4:30
RandallRandall
9,94111230
9,94111230
add a comment |
add a comment |
$begingroup$
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 SalumAlejandro Nasif Salum
4,765118
$endgroup$
add a comment |
$begingroup$
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 SalumAlejandro Nasif Salum
4,765118
$endgroup$
add a comment |
$begingroup$
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 SalumAlejandro Nasif Salum
4,765118
$endgroup$
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 SalumAlejandro Nasif Salum
4,765118
answered Dec 28 '18 at 4:31
Alejandro Nasif SalumAlejandro Nasif Salum
4,765118
answered Dec 28 '18 at 4:31
Alejandro Nasif SalumAlejandro Nasif Salum
4,765118
answered Dec 28 '18 at 4:31
Alejandro Nasif SalumAlejandro Nasif Salum
4,765118
4,765118
add a comment |
add a comment |
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1
$begingroup$
For the record, this is not the definition of a left group action, the crucial part of the definition is missing.
$endgroup$
– Did
Dec 28 '18 at 17:37