Does every subgroup of an abelian group have to be abelian? Unicorn Meta Zoo #1: Why another...
What's the difference between using dependency injection with a container and using a service locator?
Processing ADC conversion result: DMA vs Processor Registers
What is a 'Key' in computer science?
How can I wire a 9-position switch so that each position turns on one more LED than the one before?
Why did Israel vote against lifting the American embargo on Cuba?
What is the definining line between a helicopter and a drone a person can ride in?
Why isn't everyone flabbergasted about Bran's "gift"?
How to keep bees out of canned beverages?
Where can I find how to tex symbols for different fonts?
Where to find documentation for `whois` command options?
Israeli soda type drink
false 'Security alert' from Google - every login generates mails from 'no-reply@accounts.google.com'
Why did Europeans not widely domesticate foxes?
Why do people think Winterfell crypts is the safest place for women, children and old people?
How would it unbalance gameplay to rule that Weapon Master allows for picking a fighting style?
What is the numbering system used for the DSN dishes?
RIP Packet Format
What was Apollo 13's "Little Jolt" after MECO?
Why does Java have support for time zone offsets with seconds precision?
Determinant of a matrix with 2 equal rows
/bin/ls sorts differently than just ls
using NDEigensystem to solve the Mathieu equation
In search of the origins of term censor, I hit a dead end stuck with the greek term, to censor, λογοκρίνω
What is the purpose of the side handle on a hand ("eggbeater") drill?
Does every subgroup of an abelian group have to be abelian?
Unicorn Meta Zoo #1: Why another podcast?
Announcing the arrival of Valued Associate #679: Cesar ManaraA condition for a subgroup of a finitely generated free abelian group to have finite indexAdditive non-abelian group?Galois Group, Field Extension Prove AbelianCommutator subgroup is the minimal normal subgroup such that quotient group is abelianShowing that every subgroup of an abelian group is normalevery subgroup of the quaternion group is normalQuotient Group $D_2n/R$ is abelian where $R$ is the group of rotations?Proving a subgroup of a Galois group is normalNormal Subgroup of Galois GroupFaithful group action and Galois correspondence
$begingroup$
My original problem is to show that E/L is an abelian extension over L and L/F is an abelian extension over F, given that E/F is an abelian extension over F and that L is a normal extension of F such that $Fsubseteq L subseteq E$.
So far I have proved that E is a normal extension of F, E is a normal extension of L, and L is a normal extension of F. I know that to prove abelian extension I must also prove that Gal(E/L) is an abelian group. I have shown that Gal(E/L) $subseteq$ Gal (E/F). In my mind it makes sense that I cannot lose commutativity therefore my subgroup must be Abelian too. How do I show this in a proof? Is it enough to show two elements in the subgroup must also exist in the larger group and that they must be commutative in the larger group? I feel like I know what needs to be done, just not how to phrase it.
abstract-algebra group-theory galois-theory abelian-groups
edited 5 hours ago
J. W. Tanner
5,1651520
asked 6 hours ago
MT mathMT math
303
$endgroup$
add a comment |
$begingroup$
My original problem is to show that E/L is an abelian extension over L and L/F is an abelian extension over F, given that E/F is an abelian extension over F and that L is a normal extension of F such that $Fsubseteq L subseteq E$.
So far I have proved that E is a normal extension of F, E is a normal extension of L, and L is a normal extension of F. I know that to prove abelian extension I must also prove that Gal(E/L) is an abelian group. I have shown that Gal(E/L) $subseteq$ Gal (E/F). In my mind it makes sense that I cannot lose commutativity therefore my subgroup must be Abelian too. How do I show this in a proof? Is it enough to show two elements in the subgroup must also exist in the larger group and that they must be commutative in the larger group? I feel like I know what needs to be done, just not how to phrase it.
abstract-algebra group-theory galois-theory abelian-groups
edited 5 hours ago
J. W. Tanner
5,1651520
asked 6 hours ago
MT mathMT math
303
$endgroup$
add a comment |
$begingroup$
My original problem is to show that E/L is an abelian extension over L and L/F is an abelian extension over F, given that E/F is an abelian extension over F and that L is a normal extension of F such that $Fsubseteq L subseteq E$.
So far I have proved that E is a normal extension of F, E is a normal extension of L, and L is a normal extension of F. I know that to prove abelian extension I must also prove that Gal(E/L) is an abelian group. I have shown that Gal(E/L) $subseteq$ Gal (E/F). In my mind it makes sense that I cannot lose commutativity therefore my subgroup must be Abelian too. How do I show this in a proof? Is it enough to show two elements in the subgroup must also exist in the larger group and that they must be commutative in the larger group? I feel like I know what needs to be done, just not how to phrase it.
abstract-algebra group-theory galois-theory abelian-groups
edited 5 hours ago
J. W. Tanner
5,1651520
asked 6 hours ago
MT mathMT math
303
$endgroup$
My original problem is to show that E/L is an abelian extension over L and L/F is an abelian extension over F, given that E/F is an abelian extension over F and that L is a normal extension of F such that $Fsubseteq L subseteq E$.
So far I have proved that E is a normal extension of F, E is a normal extension of L, and L is a normal extension of F. I know that to prove abelian extension I must also prove that Gal(E/L) is an abelian group. I have shown that Gal(E/L) $subseteq$ Gal (E/F). In my mind it makes sense that I cannot lose commutativity therefore my subgroup must be Abelian too. How do I show this in a proof? Is it enough to show two elements in the subgroup must also exist in the larger group and that they must be commutative in the larger group? I feel like I know what needs to be done, just not how to phrase it.
abstract-algebra group-theory galois-theory abelian-groups
abstract-algebra group-theory galois-theory abelian-groups
edited 5 hours ago
J. W. Tanner
5,1651520
asked 6 hours ago
MT mathMT math
303
edited 5 hours ago
J. W. Tanner
5,1651520
asked 6 hours ago
MT mathMT math
303
edited 5 hours ago
J. W. Tanner
5,1651520
edited 5 hours ago
J. W. Tanner
5,1651520
edited 5 hours ago
J. W. Tanner
5,1651520
5,1651520
asked 6 hours ago
MT mathMT math
303
asked 6 hours ago
MT mathMT math
303
asked 6 hours ago
MT mathMT math
303
303
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
Huh, funny, we just went over this today in my algebra class.
Yes, subgroups of abelian groups are indeed abelian, and your thought process has the right idea.
Showing this is pretty easy. Take an abelian group $G$ with subgroup $H$. Then we know that, for all $a,bin H$, $ab=ba$ since it must also hold in $G$ (as $a,b in G ge H$ and $G$ is given to be abelian).
answered 6 hours ago
Eevee TrainerEevee Trainer
10.8k31843
$endgroup$
$begingroup$
can I use essentially the same reasoning to prove that L/F is an abelian extension as well?
$endgroup$
– MT math
6 hours ago
$begingroup$
I believe so, yes.
$endgroup$
– Eevee Trainer
5 hours ago
add a comment |
$begingroup$
If $G$ is an abelian group and $H$ is a subgroup, suppose $x, yin H$. Then in particular $x, yin G$, so $xy=yx$. Since $x, y$ were arbitrary, $H$ is abelian.
answered 6 hours ago
Matt SamuelMatt Samuel
39.5k63870
$endgroup$
add a comment |
Your Answer
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3198960%2fdoes-every-subgroup-of-an-abelian-group-have-to-be-abelian%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Huh, funny, we just went over this today in my algebra class.
Yes, subgroups of abelian groups are indeed abelian, and your thought process has the right idea.
Showing this is pretty easy. Take an abelian group $G$ with subgroup $H$. Then we know that, for all $a,bin H$, $ab=ba$ since it must also hold in $G$ (as $a,b in G ge H$ and $G$ is given to be abelian).
answered 6 hours ago
Eevee TrainerEevee Trainer
10.8k31843
$endgroup$
$begingroup$
can I use essentially the same reasoning to prove that L/F is an abelian extension as well?
$endgroup$
– MT math
6 hours ago
$begingroup$
I believe so, yes.
$endgroup$
– Eevee Trainer
5 hours ago
add a comment |
$begingroup$
Huh, funny, we just went over this today in my algebra class.
Yes, subgroups of abelian groups are indeed abelian, and your thought process has the right idea.
Showing this is pretty easy. Take an abelian group $G$ with subgroup $H$. Then we know that, for all $a,bin H$, $ab=ba$ since it must also hold in $G$ (as $a,b in G ge H$ and $G$ is given to be abelian).
answered 6 hours ago
Eevee TrainerEevee Trainer
10.8k31843
$endgroup$
$begingroup$
can I use essentially the same reasoning to prove that L/F is an abelian extension as well?
$endgroup$
– MT math
6 hours ago
$begingroup$
I believe so, yes.
$endgroup$
– Eevee Trainer
5 hours ago
add a comment |
$begingroup$
Huh, funny, we just went over this today in my algebra class.
Yes, subgroups of abelian groups are indeed abelian, and your thought process has the right idea.
Showing this is pretty easy. Take an abelian group $G$ with subgroup $H$. Then we know that, for all $a,bin H$, $ab=ba$ since it must also hold in $G$ (as $a,b in G ge H$ and $G$ is given to be abelian).
answered 6 hours ago
Eevee TrainerEevee Trainer
10.8k31843
$endgroup$
Huh, funny, we just went over this today in my algebra class.
Yes, subgroups of abelian groups are indeed abelian, and your thought process has the right idea.
Showing this is pretty easy. Take an abelian group $G$ with subgroup $H$. Then we know that, for all $a,bin H$, $ab=ba$ since it must also hold in $G$ (as $a,b in G ge H$ and $G$ is given to be abelian).
answered 6 hours ago
Eevee TrainerEevee Trainer
10.8k31843
answered 6 hours ago
Eevee TrainerEevee Trainer
10.8k31843
answered 6 hours ago
Eevee TrainerEevee Trainer
10.8k31843
answered 6 hours ago
Eevee TrainerEevee Trainer
10.8k31843
10.8k31843
$begingroup$
can I use essentially the same reasoning to prove that L/F is an abelian extension as well?
$endgroup$
– MT math
6 hours ago
$begingroup$
I believe so, yes.
$endgroup$
– Eevee Trainer
5 hours ago
add a comment |
$begingroup$
can I use essentially the same reasoning to prove that L/F is an abelian extension as well?
$endgroup$
– MT math
6 hours ago
$begingroup$
I believe so, yes.
$endgroup$
– Eevee Trainer
5 hours ago
$begingroup$
can I use essentially the same reasoning to prove that L/F is an abelian extension as well?
$endgroup$
– MT math
6 hours ago
$begingroup$
can I use essentially the same reasoning to prove that L/F is an abelian extension as well?
$endgroup$
– MT math
6 hours ago
$begingroup$
I believe so, yes.
$endgroup$
– Eevee Trainer
5 hours ago
$begingroup$
I believe so, yes.
$endgroup$
– Eevee Trainer
5 hours ago
add a comment |
$begingroup$
If $G$ is an abelian group and $H$ is a subgroup, suppose $x, yin H$. Then in particular $x, yin G$, so $xy=yx$. Since $x, y$ were arbitrary, $H$ is abelian.
answered 6 hours ago
Matt SamuelMatt Samuel
39.5k63870
$endgroup$
add a comment |
$begingroup$
If $G$ is an abelian group and $H$ is a subgroup, suppose $x, yin H$. Then in particular $x, yin G$, so $xy=yx$. Since $x, y$ were arbitrary, $H$ is abelian.
answered 6 hours ago
Matt SamuelMatt Samuel
39.5k63870
$endgroup$
add a comment |
$begingroup$
If $G$ is an abelian group and $H$ is a subgroup, suppose $x, yin H$. Then in particular $x, yin G$, so $xy=yx$. Since $x, y$ were arbitrary, $H$ is abelian.
answered 6 hours ago
Matt SamuelMatt Samuel
39.5k63870
$endgroup$
If $G$ is an abelian group and $H$ is a subgroup, suppose $x, yin H$. Then in particular $x, yin G$, so $xy=yx$. Since $x, y$ were arbitrary, $H$ is abelian.
answered 6 hours ago
Matt SamuelMatt Samuel
39.5k63870
answered 6 hours ago
Matt SamuelMatt Samuel
39.5k63870
answered 6 hours ago
Matt SamuelMatt Samuel
39.5k63870
answered 6 hours ago
Matt SamuelMatt Samuel
39.5k63870
39.5k63870
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3198960%2fdoes-every-subgroup-of-an-abelian-group-have-to-be-abelian%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
