A subgroup of a characteristic subgroup [closed]
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Let $G$ be a group and $A$ be a proper characteristic subgroup of $G$. If $B$ is any subgroup of $G$. Is it true that $B$ is a normal subgroup of $AB$?
abstract-algebra group-theory
asked Sep 1 at 5:28
Amrat A
1135
closed as off-topic by Derek Holt, amWhy, Adrian Keister, José Carlos Santos, Micah Sep 1 at 15:54
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." РDerek Holt, amWhy, Adrian Keister, Jos̩ Carlos Santos, Micah
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Let $G$ be a group and $A$ be a proper characteristic subgroup of $G$. If $B$ is any subgroup of $G$. Is it true that $B$ is a normal subgroup of $AB$?
abstract-algebra group-theory
asked Sep 1 at 5:28
Amrat A
1135
closed as off-topic by Derek Holt, amWhy, Adrian Keister, José Carlos Santos, Micah Sep 1 at 15:54
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." РDerek Holt, amWhy, Adrian Keister, Jos̩ Carlos Santos, Micah
1
$G$ is itself a characteristic subgroup of $G$. But $B$ is not necessarily a normal subgroup of $GB=G$.
– Alan Wang
Sep 1 at 5:33
What if $A$ is a proper subgroup of $G$?
– Amrat A
Sep 1 at 5:34
1
Consider $G = S_3$. There is a unique, hence characteristic, subgroup $A$ of order $3$. Let $B$ be any of the subgroups of order $2$. Then $B$ is not normal in $AB = G$.
– Bungo
Sep 1 at 5:37
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up vote
-1
down vote
favorite
up vote
-1
down vote
favorite
Let $G$ be a group and $A$ be a proper characteristic subgroup of $G$. If $B$ is any subgroup of $G$. Is it true that $B$ is a normal subgroup of $AB$?
abstract-algebra group-theory
asked Sep 1 at 5:28
Amrat A
1135
Let $G$ be a group and $A$ be a proper characteristic subgroup of $G$. If $B$ is any subgroup of $G$. Is it true that $B$ is a normal subgroup of $AB$?
abstract-algebra group-theory
abstract-algebra group-theory
asked Sep 1 at 5:28
Amrat A
1135
asked Sep 1 at 5:28
Amrat A
1135
edited Sep 1 at 5:35
asked Sep 1 at 5:28
Amrat A
1135
asked Sep 1 at 5:28
Amrat A
1135
asked Sep 1 at 5:28
Amrat A
1135
1135
closed as off-topic by Derek Holt, amWhy, Adrian Keister, José Carlos Santos, Micah Sep 1 at 15:54
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." РDerek Holt, amWhy, Adrian Keister, Jos̩ Carlos Santos, Micah
closed as off-topic by Derek Holt, amWhy, Adrian Keister, José Carlos Santos, Micah Sep 1 at 15:54
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." РDerek Holt, amWhy, Adrian Keister, Jos̩ Carlos Santos, Micah
1
$G$ is itself a characteristic subgroup of $G$. But $B$ is not necessarily a normal subgroup of $GB=G$.
– Alan Wang
Sep 1 at 5:33
What if $A$ is a proper subgroup of $G$?
– Amrat A
Sep 1 at 5:34
1
Consider $G = S_3$. There is a unique, hence characteristic, subgroup $A$ of order $3$. Let $B$ be any of the subgroups of order $2$. Then $B$ is not normal in $AB = G$.
– Bungo
Sep 1 at 5:37
add a comment |Â
1
$G$ is itself a characteristic subgroup of $G$. But $B$ is not necessarily a normal subgroup of $GB=G$.
– Alan Wang
Sep 1 at 5:33
What if $A$ is a proper subgroup of $G$?
– Amrat A
Sep 1 at 5:34
1
Consider $G = S_3$. There is a unique, hence characteristic, subgroup $A$ of order $3$. Let $B$ be any of the subgroups of order $2$. Then $B$ is not normal in $AB = G$.
– Bungo
Sep 1 at 5:37
1
1
$G$ is itself a characteristic subgroup of $G$. But $B$ is not necessarily a normal subgroup of $GB=G$.
– Alan Wang
Sep 1 at 5:33
$G$ is itself a characteristic subgroup of $G$. But $B$ is not necessarily a normal subgroup of $GB=G$.
– Alan Wang
Sep 1 at 5:33
What if $A$ is a proper subgroup of $G$?
– Amrat A
Sep 1 at 5:34
What if $A$ is a proper subgroup of $G$?
– Amrat A
Sep 1 at 5:34
1
1
Consider $G = S_3$. There is a unique, hence characteristic, subgroup $A$ of order $3$. Let $B$ be any of the subgroups of order $2$. Then $B$ is not normal in $AB = G$.
– Bungo
Sep 1 at 5:37
Consider $G = S_3$. There is a unique, hence characteristic, subgroup $A$ of order $3$. Let $B$ be any of the subgroups of order $2$. Then $B$ is not normal in $AB = G$.
– Bungo
Sep 1 at 5:37
add a comment |Â
1 Answer
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Consider the dihedral group $$G=langle r,s:r^3=s^2=1,sr=r^2srangle$$
Take $A=langle rrangle$ and $B=langle srangle$.
Since $A$ is the unique subgroup of order $3$, it is characteristic in $G$.
But it can be verified that $B$ is not a normal subgroup of $AB=G$ (one of the way to check is that it is not the unique Sylow $2$-subgroup of order $2$ in $G$.)
answered Sep 1 at 5:39
Alan Wang
4,584932
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
accepted
Consider the dihedral group $$G=langle r,s:r^3=s^2=1,sr=r^2srangle$$
Take $A=langle rrangle$ and $B=langle srangle$.
Since $A$ is the unique subgroup of order $3$, it is characteristic in $G$.
But it can be verified that $B$ is not a normal subgroup of $AB=G$ (one of the way to check is that it is not the unique Sylow $2$-subgroup of order $2$ in $G$.)
answered Sep 1 at 5:39
Alan Wang
4,584932
add a comment |Â
up vote
0
down vote
accepted
Consider the dihedral group $$G=langle r,s:r^3=s^2=1,sr=r^2srangle$$
Take $A=langle rrangle$ and $B=langle srangle$.
Since $A$ is the unique subgroup of order $3$, it is characteristic in $G$.
But it can be verified that $B$ is not a normal subgroup of $AB=G$ (one of the way to check is that it is not the unique Sylow $2$-subgroup of order $2$ in $G$.)
answered Sep 1 at 5:39
Alan Wang
4,584932
add a comment |Â
up vote
0
down vote
accepted
up vote
0
down vote
accepted
Consider the dihedral group $$G=langle r,s:r^3=s^2=1,sr=r^2srangle$$
Take $A=langle rrangle$ and $B=langle srangle$.
Since $A$ is the unique subgroup of order $3$, it is characteristic in $G$.
But it can be verified that $B$ is not a normal subgroup of $AB=G$ (one of the way to check is that it is not the unique Sylow $2$-subgroup of order $2$ in $G$.)
answered Sep 1 at 5:39
Alan Wang
4,584932
Consider the dihedral group $$G=langle r,s:r^3=s^2=1,sr=r^2srangle$$
Take $A=langle rrangle$ and $B=langle srangle$.
Since $A$ is the unique subgroup of order $3$, it is characteristic in $G$.
But it can be verified that $B$ is not a normal subgroup of $AB=G$ (one of the way to check is that it is not the unique Sylow $2$-subgroup of order $2$ in $G$.)
answered Sep 1 at 5:39
Alan Wang
4,584932
answered Sep 1 at 5:39
Alan Wang
4,584932
answered Sep 1 at 5:39
Alan Wang
4,584932
answered Sep 1 at 5:39
Alan Wang
4,584932
4,584932
add a comment |Â
add a comment |Â
1
$G$ is itself a characteristic subgroup of $G$. But $B$ is not necessarily a normal subgroup of $GB=G$.
– Alan Wang
Sep 1 at 5:33
What if $A$ is a proper subgroup of $G$?
– Amrat A
Sep 1 at 5:34
1
Consider $G = S_3$. There is a unique, hence characteristic, subgroup $A$ of order $3$. Let $B$ be any of the subgroups of order $2$. Then $B$ is not normal in $AB = G$.
– Bungo
Sep 1 at 5:37