Given two subgroups what are the relationes of the commutators of the two subgroups?
2
$begingroup$
Let $G$ be a group, $H$ a subgroup of $G$ and $Ntriangleleft G.$ Let $S_N$ be the set of commutators in $N$, i.e. $S_N subset N, ,aba^{-1}b^{-1}in S_N $with $ a,b in G$ and $S_H$ the set of commutators in $H$, i.e.$,S_H subset H, ,cdc^{-1}d^{-1}in S_H$ with $c, d in G.$
What holds in general, $S_N=S_H, S_Nsubset S_H$ or $S_Hsubset S_N$ in the two cases
$1.,, Hcap N={e}$, and $2.,,Hcap Nneq {e}$ ?
abstract-algebra group-theory
asked Dec 28 '18 at 11:08
user249018user249018
435138
$endgroup$
|
show 1 more comment
2
$begingroup$
Let $G$ be a group, $H$ a subgroup of $G$ and $Ntriangleleft G.$ Let $S_N$ be the set of commutators in $N$, i.e. $S_N subset N, ,aba^{-1}b^{-1}in S_N $with $ a,b in G$ and $S_H$ the set of commutators in $H$, i.e.$,S_H subset H, ,cdc^{-1}d^{-1}in S_H$ with $c, d in G.$
What holds in general, $S_N=S_H, S_Nsubset S_H$ or $S_Hsubset S_N$ in the two cases
$1.,, Hcap N={e}$, and $2.,,Hcap Nneq {e}$ ?
abstract-algebra group-theory
asked Dec 28 '18 at 11:08
user249018user249018
435138
$endgroup$
$begingroup$
How come $;a,bin G;$ for the set of commutators of $;N;$ , and like wise for $;S_H;$ ?? Or you mean the set of commutators of the whole group $;G;$ that are contained in $;N;$ (in $;H;$ , resp.) ?
$endgroup$
– DonAntonio
Dec 28 '18 at 11:19
$begingroup$
Yes, by $S_N$ I meant the set of commutators of $G$ contained in $N$ and likewise for $S_H.$ That is how it is written in an algebra book.
$endgroup$
– user249018
Dec 28 '18 at 11:30
$begingroup$
I think you can lots of test cases in the group $text{Symm}(3)timestext{Symm}(3)$.
$endgroup$
– ancientmathematician
Dec 28 '18 at 12:08
1
$begingroup$
If $Hcap N ={e}$ then the only possible commutator in both is $e$, which is indeed a commutator ($e=eee^{-1}e^{-1}$) so $S_Ncap S_H={e}$ in this case. This suggests that the second condition also will not tell you much about the relationship of the sets of commutators. If the intersection of $H$ and $N$ is small, why should there be any significant relationship here?
$endgroup$
– Mark Bennet
Dec 28 '18 at 12:29
$begingroup$
@Mark Bennet. What makes the difference if the intersection $Hcap N$ is small or not ?
$endgroup$
– user249018
Dec 28 '18 at 12:47
|
show 1 more comment
2
2
2
1
$begingroup$
Let $G$ be a group, $H$ a subgroup of $G$ and $Ntriangleleft G.$ Let $S_N$ be the set of commutators in $N$, i.e. $S_N subset N, ,aba^{-1}b^{-1}in S_N $with $ a,b in G$ and $S_H$ the set of commutators in $H$, i.e.$,S_H subset H, ,cdc^{-1}d^{-1}in S_H$ with $c, d in G.$
What holds in general, $S_N=S_H, S_Nsubset S_H$ or $S_Hsubset S_N$ in the two cases
$1.,, Hcap N={e}$, and $2.,,Hcap Nneq {e}$ ?
abstract-algebra group-theory
asked Dec 28 '18 at 11:08
user249018user249018
435138
$endgroup$
Let $G$ be a group, $H$ a subgroup of $G$ and $Ntriangleleft G.$ Let $S_N$ be the set of commutators in $N$, i.e. $S_N subset N, ,aba^{-1}b^{-1}in S_N $with $ a,b in G$ and $S_H$ the set of commutators in $H$, i.e.$,S_H subset H, ,cdc^{-1}d^{-1}in S_H$ with $c, d in G.$
What holds in general, $S_N=S_H, S_Nsubset S_H$ or $S_Hsubset S_N$ in the two cases
$1.,, Hcap N={e}$, and $2.,,Hcap Nneq {e}$ ?
abstract-algebra group-theory
abstract-algebra group-theory
asked Dec 28 '18 at 11:08
user249018user249018
435138
asked Dec 28 '18 at 11:08
user249018user249018
435138
asked Dec 28 '18 at 11:08
user249018user249018
435138
asked Dec 28 '18 at 11:08
user249018user249018
435138
asked Dec 28 '18 at 11:08
user249018user249018
435138
435138
$begingroup$
How come $;a,bin G;$ for the set of commutators of $;N;$ , and like wise for $;S_H;$ ?? Or you mean the set of commutators of the whole group $;G;$ that are contained in $;N;$ (in $;H;$ , resp.) ?
$endgroup$
– DonAntonio
Dec 28 '18 at 11:19
$begingroup$
Yes, by $S_N$ I meant the set of commutators of $G$ contained in $N$ and likewise for $S_H.$ That is how it is written in an algebra book.
$endgroup$
– user249018
Dec 28 '18 at 11:30
$begingroup$
I think you can lots of test cases in the group $text{Symm}(3)timestext{Symm}(3)$.
$endgroup$
– ancientmathematician
Dec 28 '18 at 12:08
1
$begingroup$
If $Hcap N ={e}$ then the only possible commutator in both is $e$, which is indeed a commutator ($e=eee^{-1}e^{-1}$) so $S_Ncap S_H={e}$ in this case. This suggests that the second condition also will not tell you much about the relationship of the sets of commutators. If the intersection of $H$ and $N$ is small, why should there be any significant relationship here?
$endgroup$
– Mark Bennet
Dec 28 '18 at 12:29
$begingroup$
@Mark Bennet. What makes the difference if the intersection $Hcap N$ is small or not ?
$endgroup$
– user249018
Dec 28 '18 at 12:47
|
show 1 more comment
$begingroup$
How come $;a,bin G;$ for the set of commutators of $;N;$ , and like wise for $;S_H;$ ?? Or you mean the set of commutators of the whole group $;G;$ that are contained in $;N;$ (in $;H;$ , resp.) ?
$endgroup$
– DonAntonio
Dec 28 '18 at 11:19
$begingroup$
Yes, by $S_N$ I meant the set of commutators of $G$ contained in $N$ and likewise for $S_H.$ That is how it is written in an algebra book.
$endgroup$
– user249018
Dec 28 '18 at 11:30
$begingroup$
I think you can lots of test cases in the group $text{Symm}(3)timestext{Symm}(3)$.
$endgroup$
– ancientmathematician
Dec 28 '18 at 12:08
1
$begingroup$
If $Hcap N ={e}$ then the only possible commutator in both is $e$, which is indeed a commutator ($e=eee^{-1}e^{-1}$) so $S_Ncap S_H={e}$ in this case. This suggests that the second condition also will not tell you much about the relationship of the sets of commutators. If the intersection of $H$ and $N$ is small, why should there be any significant relationship here?
$endgroup$
– Mark Bennet
Dec 28 '18 at 12:29
$begingroup$
@Mark Bennet. What makes the difference if the intersection $Hcap N$ is small or not ?
$endgroup$
– user249018
Dec 28 '18 at 12:47
$begingroup$
How come $;a,bin G;$ for the set of commutators of $;N;$ , and like wise for $;S_H;$ ?? Or you mean the set of commutators of the whole group $;G;$ that are contained in $;N;$ (in $;H;$ , resp.) ?
$endgroup$
– DonAntonio
Dec 28 '18 at 11:19
$begingroup$
How come $;a,bin G;$ for the set of commutators of $;N;$ , and like wise for $;S_H;$ ?? Or you mean the set of commutators of the whole group $;G;$ that are contained in $;N;$ (in $;H;$ , resp.) ?
$endgroup$
– DonAntonio
Dec 28 '18 at 11:19
$begingroup$
Yes, by $S_N$ I meant the set of commutators of $G$ contained in $N$ and likewise for $S_H.$ That is how it is written in an algebra book.
$endgroup$
– user249018
Dec 28 '18 at 11:30
$begingroup$
Yes, by $S_N$ I meant the set of commutators of $G$ contained in $N$ and likewise for $S_H.$ That is how it is written in an algebra book.
$endgroup$
– user249018
Dec 28 '18 at 11:30
$begingroup$
I think you can lots of test cases in the group $text{Symm}(3)timestext{Symm}(3)$.
$endgroup$
– ancientmathematician
Dec 28 '18 at 12:08
$begingroup$
I think you can lots of test cases in the group $text{Symm}(3)timestext{Symm}(3)$.
$endgroup$
– ancientmathematician
Dec 28 '18 at 12:08
1
1
$begingroup$
If $Hcap N ={e}$ then the only possible commutator in both is $e$, which is indeed a commutator ($e=eee^{-1}e^{-1}$) so $S_Ncap S_H={e}$ in this case. This suggests that the second condition also will not tell you much about the relationship of the sets of commutators. If the intersection of $H$ and $N$ is small, why should there be any significant relationship here?
$endgroup$
– Mark Bennet
Dec 28 '18 at 12:29
$begingroup$
If $Hcap N ={e}$ then the only possible commutator in both is $e$, which is indeed a commutator ($e=eee^{-1}e^{-1}$) so $S_Ncap S_H={e}$ in this case. This suggests that the second condition also will not tell you much about the relationship of the sets of commutators. If the intersection of $H$ and $N$ is small, why should there be any significant relationship here?
$endgroup$
– Mark Bennet
Dec 28 '18 at 12:29
$begingroup$
@Mark Bennet. What makes the difference if the intersection $Hcap N$ is small or not ?
$endgroup$
– user249018
Dec 28 '18 at 12:47
$begingroup$
@Mark Bennet. What makes the difference if the intersection $Hcap N$ is small or not ?
$endgroup$
– user249018
Dec 28 '18 at 12:47
|
show 1 more comment
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$begingroup$
How come $;a,bin G;$ for the set of commutators of $;N;$ , and like wise for $;S_H;$ ?? Or you mean the set of commutators of the whole group $;G;$ that are contained in $;N;$ (in $;H;$ , resp.) ?
$endgroup$
– DonAntonio
Dec 28 '18 at 11:19
$begingroup$
Yes, by $S_N$ I meant the set of commutators of $G$ contained in $N$ and likewise for $S_H.$ That is how it is written in an algebra book.
$endgroup$
– user249018
Dec 28 '18 at 11:30
$begingroup$
I think you can lots of test cases in the group $text{Symm}(3)timestext{Symm}(3)$.
$endgroup$
– ancientmathematician
Dec 28 '18 at 12:08
1
$begingroup$
If $Hcap N ={e}$ then the only possible commutator in both is $e$, which is indeed a commutator ($e=eee^{-1}e^{-1}$) so $S_Ncap S_H={e}$ in this case. This suggests that the second condition also will not tell you much about the relationship of the sets of commutators. If the intersection of $H$ and $N$ is small, why should there be any significant relationship here?
$endgroup$
– Mark Bennet
Dec 28 '18 at 12:29
$begingroup$
@Mark Bennet. What makes the difference if the intersection $Hcap N$ is small or not ?
$endgroup$
– user249018
Dec 28 '18 at 12:47