Number of subgroups of $S_6$ isomorphic to $C_3times C_3$
I have a doubt, I do not know if my method of solving this problem is correct. I have to count how many isomorphic subgroups to $ mathbb{Z}_3times mathbb{Z}_3$ there are in $ S_6 $. being abelian I have to find two generators that switch between them, so I think you can choose a three cycle and then I have 40 choices, having the second generator forced (the other three cycle, and its power) then in all 80 choices, dividing by the number of elements of order 3 I have that ultimately there are 20 in the subgroups searched for.
group-theory group-isomorphism
edited Nov 25 at 20:10
the_fox
2,34911431
asked Nov 25 at 19:45
Cristian Sopio
112
|
show 1 more comment
I have a doubt, I do not know if my method of solving this problem is correct. I have to count how many isomorphic subgroups to $ mathbb{Z}_3times mathbb{Z}_3$ there are in $ S_6 $. being abelian I have to find two generators that switch between them, so I think you can choose a three cycle and then I have 40 choices, having the second generator forced (the other three cycle, and its power) then in all 80 choices, dividing by the number of elements of order 3 I have that ultimately there are 20 in the subgroups searched for.
group-theory group-isomorphism
edited Nov 25 at 20:10
the_fox
2,34911431
asked Nov 25 at 19:45
Cristian Sopio
112
Write$times$for $times$ and either$mathbb{Z}$or$Bbb{Z}$for $Bbb Z$. To find out more, search online for a MathJax tutorial.
– Shaun
Nov 25 at 19:50
Please edit the question.
– Shaun
Nov 25 at 19:55
You have chosen each subgroup twice.
– Derek Holt
Nov 25 at 19:57
Why I choose twice the subgroup?
– Cristian Sopio
Nov 25 at 20:02
2
$langle (123), (456) rangle = langle (456), (123) rangle$
– the_fox
Nov 25 at 20:09
|
show 1 more comment
I have a doubt, I do not know if my method of solving this problem is correct. I have to count how many isomorphic subgroups to $ mathbb{Z}_3times mathbb{Z}_3$ there are in $ S_6 $. being abelian I have to find two generators that switch between them, so I think you can choose a three cycle and then I have 40 choices, having the second generator forced (the other three cycle, and its power) then in all 80 choices, dividing by the number of elements of order 3 I have that ultimately there are 20 in the subgroups searched for.
group-theory group-isomorphism
edited Nov 25 at 20:10
the_fox
2,34911431
asked Nov 25 at 19:45
Cristian Sopio
112
I have a doubt, I do not know if my method of solving this problem is correct. I have to count how many isomorphic subgroups to $ mathbb{Z}_3times mathbb{Z}_3$ there are in $ S_6 $. being abelian I have to find two generators that switch between them, so I think you can choose a three cycle and then I have 40 choices, having the second generator forced (the other three cycle, and its power) then in all 80 choices, dividing by the number of elements of order 3 I have that ultimately there are 20 in the subgroups searched for.
group-theory group-isomorphism
group-theory group-isomorphism
edited Nov 25 at 20:10
the_fox
2,34911431
asked Nov 25 at 19:45
Cristian Sopio
112
edited Nov 25 at 20:10
the_fox
2,34911431
asked Nov 25 at 19:45
Cristian Sopio
112
edited Nov 25 at 20:10
the_fox
2,34911431
edited Nov 25 at 20:10
the_fox
2,34911431
edited Nov 25 at 20:10
the_fox
2,34911431
2,34911431
asked Nov 25 at 19:45
Cristian Sopio
112
asked Nov 25 at 19:45
Cristian Sopio
112
asked Nov 25 at 19:45
Cristian Sopio
112
112
Write$times$for $times$ and either$mathbb{Z}$or$Bbb{Z}$for $Bbb Z$. To find out more, search online for a MathJax tutorial.
– Shaun
Nov 25 at 19:50
Please edit the question.
– Shaun
Nov 25 at 19:55
You have chosen each subgroup twice.
– Derek Holt
Nov 25 at 19:57
Why I choose twice the subgroup?
– Cristian Sopio
Nov 25 at 20:02
2
$langle (123), (456) rangle = langle (456), (123) rangle$
– the_fox
Nov 25 at 20:09
|
show 1 more comment
Write$times$for $times$ and either$mathbb{Z}$or$Bbb{Z}$for $Bbb Z$. To find out more, search online for a MathJax tutorial.
– Shaun
Nov 25 at 19:50
Please edit the question.
– Shaun
Nov 25 at 19:55
You have chosen each subgroup twice.
– Derek Holt
Nov 25 at 19:57
Why I choose twice the subgroup?
– Cristian Sopio
Nov 25 at 20:02
2
$langle (123), (456) rangle = langle (456), (123) rangle$
– the_fox
Nov 25 at 20:09
Write
$times$ for $times$ and either $mathbb{Z}$ or $Bbb{Z}$ for $Bbb Z$. To find out more, search online for a MathJax tutorial.– Shaun
Nov 25 at 19:50
Write
$times$ for $times$ and either $mathbb{Z}$ or $Bbb{Z}$ for $Bbb Z$. To find out more, search online for a MathJax tutorial.– Shaun
Nov 25 at 19:50
Please edit the question.
– Shaun
Nov 25 at 19:55
Please edit the question.
– Shaun
Nov 25 at 19:55
You have chosen each subgroup twice.
– Derek Holt
Nov 25 at 19:57
You have chosen each subgroup twice.
– Derek Holt
Nov 25 at 19:57
Why I choose twice the subgroup?
– Cristian Sopio
Nov 25 at 20:02
Why I choose twice the subgroup?
– Cristian Sopio
Nov 25 at 20:02
2
2
$langle (123), (456) rangle = langle (456), (123) rangle$
– the_fox
Nov 25 at 20:09
$langle (123), (456) rangle = langle (456), (123) rangle$
– the_fox
Nov 25 at 20:09
|
show 1 more comment
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Write
$times$for $times$ and either$mathbb{Z}$or$Bbb{Z}$for $Bbb Z$. To find out more, search online for a MathJax tutorial.– Shaun
Nov 25 at 19:50
Please edit the question.
– Shaun
Nov 25 at 19:55
You have chosen each subgroup twice.
– Derek Holt
Nov 25 at 19:57
Why I choose twice the subgroup?
– Cristian Sopio
Nov 25 at 20:02
2
$langle (123), (456) rangle = langle (456), (123) rangle$
– the_fox
Nov 25 at 20:09