Let $G$ be a group and $H, H'$ be subgroups of $G$ where $H$ is normal. Under which circumstances is $H cap...
0
To be more specific, what kind of assumptions do I have to make for $H'$ to obtain this assertion? Which one are necessary and which one are sufficient?
For instance, we could say $H subset H'$, but this case seems rather boring. Do you have some better ideas? Thank you!
abstract-algebra group-theory normal-subgroups quotient-group
asked Nov 25 at 19:37
Diglett
879520
|
show 1 more comment
0
To be more specific, what kind of assumptions do I have to make for $H'$ to obtain this assertion? Which one are necessary and which one are sufficient?
For instance, we could say $H subset H'$, but this case seems rather boring. Do you have some better ideas? Thank you!
abstract-algebra group-theory normal-subgroups quotient-group
asked Nov 25 at 19:37
Diglett
879520
1
What else do you expect? In order to be normal subgroup, it would need to be subgroup first. Once it is a subgroup, it automatically is normal since it is normal in a bigger group.
– Quang Hoang
Nov 25 at 19:41
2
If $N$ is normal in $G$ and $K$ is a subgroup of $G$ then $N cap K$ is a normal subgroup of $K$. You don't need any assumptions on $K$ for that to be true.
– the_fox
Nov 25 at 19:48
I just noticed that I made a huge mistake in the headline: $H'$ does not has to be necessarily normal!
– Diglett
Nov 25 at 20:47
1
In that case, I don't see what difference $H$ being normal in $G$ makes. Anyway, you'll find that with questions like this you only get what you seek in "trivial" cases.
– the_fox
Nov 25 at 21:03
1
Please include the full text of your query in the body of your message, not just in the title. In fact, the title should be suggestive, but should not be a full two sentences!
– Arturo Magidin
Nov 25 at 22:50
|
show 1 more comment
0
0
0
To be more specific, what kind of assumptions do I have to make for $H'$ to obtain this assertion? Which one are necessary and which one are sufficient?
For instance, we could say $H subset H'$, but this case seems rather boring. Do you have some better ideas? Thank you!
abstract-algebra group-theory normal-subgroups quotient-group
asked Nov 25 at 19:37
Diglett
879520
To be more specific, what kind of assumptions do I have to make for $H'$ to obtain this assertion? Which one are necessary and which one are sufficient?
For instance, we could say $H subset H'$, but this case seems rather boring. Do you have some better ideas? Thank you!
abstract-algebra group-theory normal-subgroups quotient-group
abstract-algebra group-theory normal-subgroups quotient-group
asked Nov 25 at 19:37
Diglett
879520
asked Nov 25 at 19:37
Diglett
879520
edited Nov 25 at 20:46
asked Nov 25 at 19:37
Diglett
879520
asked Nov 25 at 19:37
Diglett
879520
asked Nov 25 at 19:37
Diglett
879520
879520
1
What else do you expect? In order to be normal subgroup, it would need to be subgroup first. Once it is a subgroup, it automatically is normal since it is normal in a bigger group.
– Quang Hoang
Nov 25 at 19:41
2
If $N$ is normal in $G$ and $K$ is a subgroup of $G$ then $N cap K$ is a normal subgroup of $K$. You don't need any assumptions on $K$ for that to be true.
– the_fox
Nov 25 at 19:48
I just noticed that I made a huge mistake in the headline: $H'$ does not has to be necessarily normal!
– Diglett
Nov 25 at 20:47
1
In that case, I don't see what difference $H$ being normal in $G$ makes. Anyway, you'll find that with questions like this you only get what you seek in "trivial" cases.
– the_fox
Nov 25 at 21:03
1
Please include the full text of your query in the body of your message, not just in the title. In fact, the title should be suggestive, but should not be a full two sentences!
– Arturo Magidin
Nov 25 at 22:50
|
show 1 more comment
1
What else do you expect? In order to be normal subgroup, it would need to be subgroup first. Once it is a subgroup, it automatically is normal since it is normal in a bigger group.
– Quang Hoang
Nov 25 at 19:41
2
If $N$ is normal in $G$ and $K$ is a subgroup of $G$ then $N cap K$ is a normal subgroup of $K$. You don't need any assumptions on $K$ for that to be true.
– the_fox
Nov 25 at 19:48
I just noticed that I made a huge mistake in the headline: $H'$ does not has to be necessarily normal!
– Diglett
Nov 25 at 20:47
1
In that case, I don't see what difference $H$ being normal in $G$ makes. Anyway, you'll find that with questions like this you only get what you seek in "trivial" cases.
– the_fox
Nov 25 at 21:03
1
Please include the full text of your query in the body of your message, not just in the title. In fact, the title should be suggestive, but should not be a full two sentences!
– Arturo Magidin
Nov 25 at 22:50
1
1
What else do you expect? In order to be normal subgroup, it would need to be subgroup first. Once it is a subgroup, it automatically is normal since it is normal in a bigger group.
– Quang Hoang
Nov 25 at 19:41
What else do you expect? In order to be normal subgroup, it would need to be subgroup first. Once it is a subgroup, it automatically is normal since it is normal in a bigger group.
– Quang Hoang
Nov 25 at 19:41
2
2
If $N$ is normal in $G$ and $K$ is a subgroup of $G$ then $N cap K$ is a normal subgroup of $K$. You don't need any assumptions on $K$ for that to be true.
– the_fox
Nov 25 at 19:48
If $N$ is normal in $G$ and $K$ is a subgroup of $G$ then $N cap K$ is a normal subgroup of $K$. You don't need any assumptions on $K$ for that to be true.
– the_fox
Nov 25 at 19:48
I just noticed that I made a huge mistake in the headline: $H'$ does not has to be necessarily normal!
– Diglett
Nov 25 at 20:47
I just noticed that I made a huge mistake in the headline: $H'$ does not has to be necessarily normal!
– Diglett
Nov 25 at 20:47
1
1
In that case, I don't see what difference $H$ being normal in $G$ makes. Anyway, you'll find that with questions like this you only get what you seek in "trivial" cases.
– the_fox
Nov 25 at 21:03
In that case, I don't see what difference $H$ being normal in $G$ makes. Anyway, you'll find that with questions like this you only get what you seek in "trivial" cases.
– the_fox
Nov 25 at 21:03
1
1
Please include the full text of your query in the body of your message, not just in the title. In fact, the title should be suggestive, but should not be a full two sentences!
– Arturo Magidin
Nov 25 at 22:50
Please include the full text of your query in the body of your message, not just in the title. In fact, the title should be suggestive, but should not be a full two sentences!
– Arturo Magidin
Nov 25 at 22:50
|
show 1 more comment
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1
What else do you expect? In order to be normal subgroup, it would need to be subgroup first. Once it is a subgroup, it automatically is normal since it is normal in a bigger group.
– Quang Hoang
Nov 25 at 19:41
2
If $N$ is normal in $G$ and $K$ is a subgroup of $G$ then $N cap K$ is a normal subgroup of $K$. You don't need any assumptions on $K$ for that to be true.
– the_fox
Nov 25 at 19:48
I just noticed that I made a huge mistake in the headline: $H'$ does not has to be necessarily normal!
– Diglett
Nov 25 at 20:47
1
In that case, I don't see what difference $H$ being normal in $G$ makes. Anyway, you'll find that with questions like this you only get what you seek in "trivial" cases.
– the_fox
Nov 25 at 21:03
1
Please include the full text of your query in the body of your message, not just in the title. In fact, the title should be suggestive, but should not be a full two sentences!
– Arturo Magidin
Nov 25 at 22:50