Let $H$ be a subgroup of $G$. Show that if $G/H$ is abelian, then $ghg^{-1}h^{-1}$ is in $H$ for all $g, h$...
Let $H$ be a subgroup of $G$. Show that if $G/H$ is abelian, then $ghg^{-1}h^{-1}$ is in $H$ for all $g, h$ in $G$..
I know if $G/H$ is abelian, $(g_1H)(g_2H)=(g_2H)(g_1H)$. But I don't know how to approach from here.
abstract-algebra
edited Nov 20 '18 at 11:24
1ENİGMA1
958416
asked Nov 19 '18 at 4:34
david D
875
add a comment |
Let $H$ be a subgroup of $G$. Show that if $G/H$ is abelian, then $ghg^{-1}h^{-1}$ is in $H$ for all $g, h$ in $G$..
I know if $G/H$ is abelian, $(g_1H)(g_2H)=(g_2H)(g_1H)$. But I don't know how to approach from here.
abstract-algebra
edited Nov 20 '18 at 11:24
1ENİGMA1
958416
asked Nov 19 '18 at 4:34
david D
875
add a comment |
Let $H$ be a subgroup of $G$. Show that if $G/H$ is abelian, then $ghg^{-1}h^{-1}$ is in $H$ for all $g, h$ in $G$..
I know if $G/H$ is abelian, $(g_1H)(g_2H)=(g_2H)(g_1H)$. But I don't know how to approach from here.
abstract-algebra
edited Nov 20 '18 at 11:24
1ENİGMA1
958416
asked Nov 19 '18 at 4:34
david D
875
Let $H$ be a subgroup of $G$. Show that if $G/H$ is abelian, then $ghg^{-1}h^{-1}$ is in $H$ for all $g, h$ in $G$..
I know if $G/H$ is abelian, $(g_1H)(g_2H)=(g_2H)(g_1H)$. But I don't know how to approach from here.
abstract-algebra
abstract-algebra
edited Nov 20 '18 at 11:24
1ENİGMA1
958416
asked Nov 19 '18 at 4:34
david D
875
edited Nov 20 '18 at 11:24
1ENİGMA1
958416
asked Nov 19 '18 at 4:34
david D
875
edited Nov 20 '18 at 11:24
1ENİGMA1
958416
edited Nov 20 '18 at 11:24
1ENİGMA1
958416
edited Nov 20 '18 at 11:24
1ENİGMA1
958416
958416
asked Nov 19 '18 at 4:34
david D
875
asked Nov 19 '18 at 4:34
david D
875
asked Nov 19 '18 at 4:34
david D
875
875
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
You have it. So what is $(g_1g_2g_1^{-1}g_2^{-1})H$? (P.S. Probably it's better not to use the letter $h$ for an element of $G$; it suggests an element of $H$. So I think your switching to $g_1$ and $g_2$ was preferable.)
answered Nov 19 '18 at 4:44
Ted Shifrin
62.9k44489
Sorry I am still confused, why (g1g2g1−1g2−1)H can prove the statement?
– david D
Nov 19 '18 at 4:56
1
If that element of $G/H$ equals $eH$, then $g_1g_2g_1^{-1}g_2^{-1}in H$.
– Ted Shifrin
Nov 19 '18 at 4:59
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
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%2f3004526%2flet-h-be-a-subgroup-of-g-show-that-if-g-h-is-abelian-then-ghg-1h-1%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
You have it. So what is $(g_1g_2g_1^{-1}g_2^{-1})H$? (P.S. Probably it's better not to use the letter $h$ for an element of $G$; it suggests an element of $H$. So I think your switching to $g_1$ and $g_2$ was preferable.)
answered Nov 19 '18 at 4:44
Ted Shifrin
62.9k44489
Sorry I am still confused, why (g1g2g1−1g2−1)H can prove the statement?
– david D
Nov 19 '18 at 4:56
1
If that element of $G/H$ equals $eH$, then $g_1g_2g_1^{-1}g_2^{-1}in H$.
– Ted Shifrin
Nov 19 '18 at 4:59
add a comment |
You have it. So what is $(g_1g_2g_1^{-1}g_2^{-1})H$? (P.S. Probably it's better not to use the letter $h$ for an element of $G$; it suggests an element of $H$. So I think your switching to $g_1$ and $g_2$ was preferable.)
answered Nov 19 '18 at 4:44
Ted Shifrin
62.9k44489
Sorry I am still confused, why (g1g2g1−1g2−1)H can prove the statement?
– david D
Nov 19 '18 at 4:56
1
If that element of $G/H$ equals $eH$, then $g_1g_2g_1^{-1}g_2^{-1}in H$.
– Ted Shifrin
Nov 19 '18 at 4:59
add a comment |
You have it. So what is $(g_1g_2g_1^{-1}g_2^{-1})H$? (P.S. Probably it's better not to use the letter $h$ for an element of $G$; it suggests an element of $H$. So I think your switching to $g_1$ and $g_2$ was preferable.)
answered Nov 19 '18 at 4:44
Ted Shifrin
62.9k44489
You have it. So what is $(g_1g_2g_1^{-1}g_2^{-1})H$? (P.S. Probably it's better not to use the letter $h$ for an element of $G$; it suggests an element of $H$. So I think your switching to $g_1$ and $g_2$ was preferable.)
answered Nov 19 '18 at 4:44
Ted Shifrin
62.9k44489
answered Nov 19 '18 at 4:44
Ted Shifrin
62.9k44489
answered Nov 19 '18 at 4:44
Ted Shifrin
62.9k44489
answered Nov 19 '18 at 4:44
Ted Shifrin
62.9k44489
62.9k44489
Sorry I am still confused, why (g1g2g1−1g2−1)H can prove the statement?
– david D
Nov 19 '18 at 4:56
1
If that element of $G/H$ equals $eH$, then $g_1g_2g_1^{-1}g_2^{-1}in H$.
– Ted Shifrin
Nov 19 '18 at 4:59
add a comment |
Sorry I am still confused, why (g1g2g1−1g2−1)H can prove the statement?
– david D
Nov 19 '18 at 4:56
1
If that element of $G/H$ equals $eH$, then $g_1g_2g_1^{-1}g_2^{-1}in H$.
– Ted Shifrin
Nov 19 '18 at 4:59
Sorry I am still confused, why (g1g2g1−1g2−1)H can prove the statement?
– david D
Nov 19 '18 at 4:56
Sorry I am still confused, why (g1g2g1−1g2−1)H can prove the statement?
– david D
Nov 19 '18 at 4:56
1
1
If that element of $G/H$ equals $eH$, then $g_1g_2g_1^{-1}g_2^{-1}in H$.
– Ted Shifrin
Nov 19 '18 at 4:59
If that element of $G/H$ equals $eH$, then $g_1g_2g_1^{-1}g_2^{-1}in H$.
– Ted Shifrin
Nov 19 '18 at 4:59
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.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- 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.
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%2f3004526%2flet-h-be-a-subgroup-of-g-show-that-if-g-h-is-abelian-then-ghg-1h-1%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
