Proof related to number of conjugates of an element of a group
Clash Royale CLAN TAG#URR8PPP
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Theorem
Let G be a finite group and let a be an element of G . Then,$ |cl(a)|=|G:C(a)|.$
Attempt
Consider the function$T$ that sends the coset $xC(a)$ to the conjugate $xax^-1$.
$xC(a)=yC(a)$ which implies $y^
-1x in C(a)$
which inturn implies $y^-1xa=ay^-1x$.Rearranging we get $xax^-1=yay^-1$. So T is well defined.
By reversing the argument ,we can see T is one-one. Clearly T is onto.
Doubt
Is my proof correct?
How to prove T preserves operation? Or does it preserves operation. (It is not required for the proof)
abstract-algebra group-theory proof-verification
asked Sep 2 at 5:58
blue boy
1,117513
add a comment |Â
up vote
1
down vote
favorite
Theorem
Let G be a finite group and let a be an element of G . Then,$ |cl(a)|=|G:C(a)|.$
Attempt
Consider the function$T$ that sends the coset $xC(a)$ to the conjugate $xax^-1$.
$xC(a)=yC(a)$ which implies $y^
-1x in C(a)$
which inturn implies $y^-1xa=ay^-1x$.Rearranging we get $xax^-1=yay^-1$. So T is well defined.
By reversing the argument ,we can see T is one-one. Clearly T is onto.
Doubt
Is my proof correct?
How to prove T preserves operation? Or does it preserves operation. (It is not required for the proof)
abstract-algebra group-theory proof-verification
asked Sep 2 at 5:58
blue boy
1,117513
Why do you need $T$ to preserve an operation? All you need is that it's a bijection in order to conclude that the cardinalities of the two sets are equal.
– Bungo
Sep 2 at 6:02
My bad. I have edited the question.
– blue boy
Sep 2 at 6:03
3
Your proof is fine as written. Note that in general, $cl(a)$ and $G / C(a)$ aren't even groups, so it wouldn't make a lot of sense to ask whether $T$ preserves the group operation.
– Bungo
Sep 2 at 6:05
If $C(a)$ is replaced by center $Z(G)$ then the operation will be preserved right?
– blue boy
Sep 2 at 6:25
$T:gZ(G) to phi_g $where$ phi_g= gxg^-1$
– blue boy
Sep 2 at 7:03
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Theorem
Let G be a finite group and let a be an element of G . Then,$ |cl(a)|=|G:C(a)|.$
Attempt
Consider the function$T$ that sends the coset $xC(a)$ to the conjugate $xax^-1$.
$xC(a)=yC(a)$ which implies $y^
-1x in C(a)$
which inturn implies $y^-1xa=ay^-1x$.Rearranging we get $xax^-1=yay^-1$. So T is well defined.
By reversing the argument ,we can see T is one-one. Clearly T is onto.
Doubt
Is my proof correct?
How to prove T preserves operation? Or does it preserves operation. (It is not required for the proof)
abstract-algebra group-theory proof-verification
asked Sep 2 at 5:58
blue boy
1,117513
Theorem
Let G be a finite group and let a be an element of G . Then,$ |cl(a)|=|G:C(a)|.$
Attempt
Consider the function$T$ that sends the coset $xC(a)$ to the conjugate $xax^-1$.
$xC(a)=yC(a)$ which implies $y^
-1x in C(a)$
which inturn implies $y^-1xa=ay^-1x$.Rearranging we get $xax^-1=yay^-1$. So T is well defined.
By reversing the argument ,we can see T is one-one. Clearly T is onto.
Doubt
Is my proof correct?
How to prove T preserves operation? Or does it preserves operation. (It is not required for the proof)
abstract-algebra group-theory proof-verification
abstract-algebra group-theory proof-verification
asked Sep 2 at 5:58
blue boy
1,117513
asked Sep 2 at 5:58
blue boy
1,117513
asked Sep 2 at 5:58
blue boy
1,117513
asked Sep 2 at 5:58
blue boy
1,117513
asked Sep 2 at 5:58
blue boy
1,117513
1,117513
Why do you need $T$ to preserve an operation? All you need is that it's a bijection in order to conclude that the cardinalities of the two sets are equal.
– Bungo
Sep 2 at 6:02
My bad. I have edited the question.
– blue boy
Sep 2 at 6:03
3
Your proof is fine as written. Note that in general, $cl(a)$ and $G / C(a)$ aren't even groups, so it wouldn't make a lot of sense to ask whether $T$ preserves the group operation.
– Bungo
Sep 2 at 6:05
If $C(a)$ is replaced by center $Z(G)$ then the operation will be preserved right?
– blue boy
Sep 2 at 6:25
$T:gZ(G) to phi_g $where$ phi_g= gxg^-1$
– blue boy
Sep 2 at 7:03
add a comment |Â
Why do you need $T$ to preserve an operation? All you need is that it's a bijection in order to conclude that the cardinalities of the two sets are equal.
– Bungo
Sep 2 at 6:02
My bad. I have edited the question.
– blue boy
Sep 2 at 6:03
3
Your proof is fine as written. Note that in general, $cl(a)$ and $G / C(a)$ aren't even groups, so it wouldn't make a lot of sense to ask whether $T$ preserves the group operation.
– Bungo
Sep 2 at 6:05
If $C(a)$ is replaced by center $Z(G)$ then the operation will be preserved right?
– blue boy
Sep 2 at 6:25
$T:gZ(G) to phi_g $where$ phi_g= gxg^-1$
– blue boy
Sep 2 at 7:03
Why do you need $T$ to preserve an operation? All you need is that it's a bijection in order to conclude that the cardinalities of the two sets are equal.
– Bungo
Sep 2 at 6:02
Why do you need $T$ to preserve an operation? All you need is that it's a bijection in order to conclude that the cardinalities of the two sets are equal.
– Bungo
Sep 2 at 6:02
My bad. I have edited the question.
– blue boy
Sep 2 at 6:03
My bad. I have edited the question.
– blue boy
Sep 2 at 6:03
3
3
Your proof is fine as written. Note that in general, $cl(a)$ and $G / C(a)$ aren't even groups, so it wouldn't make a lot of sense to ask whether $T$ preserves the group operation.
– Bungo
Sep 2 at 6:05
Your proof is fine as written. Note that in general, $cl(a)$ and $G / C(a)$ aren't even groups, so it wouldn't make a lot of sense to ask whether $T$ preserves the group operation.
– Bungo
Sep 2 at 6:05
If $C(a)$ is replaced by center $Z(G)$ then the operation will be preserved right?
– blue boy
Sep 2 at 6:25
If $C(a)$ is replaced by center $Z(G)$ then the operation will be preserved right?
– blue boy
Sep 2 at 6:25
$T:gZ(G) to phi_g $where$ phi_g= gxg^-1$
– blue boy
Sep 2 at 7:03
$T:gZ(G) to phi_g $where$ phi_g= gxg^-1$
– blue boy
Sep 2 at 7:03
add a comment |Â
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Why do you need $T$ to preserve an operation? All you need is that it's a bijection in order to conclude that the cardinalities of the two sets are equal.
– Bungo
Sep 2 at 6:02
My bad. I have edited the question.
– blue boy
Sep 2 at 6:03
3
Your proof is fine as written. Note that in general, $cl(a)$ and $G / C(a)$ aren't even groups, so it wouldn't make a lot of sense to ask whether $T$ preserves the group operation.
– Bungo
Sep 2 at 6:05
If $C(a)$ is replaced by center $Z(G)$ then the operation will be preserved right?
– blue boy
Sep 2 at 6:25
$T:gZ(G) to phi_g $where$ phi_g= gxg^-1$
– blue boy
Sep 2 at 7:03