For a subgroup $H$ of a finite group $G$ , when does $lvert operatorname{Aut}(H)rvert$ divide $lvert...
$begingroup$
Let $H$ be a subgroup of a finite group $G$. Is it true that $lvert operatorname{Aut}(H)rvert$ divides $lvert operatorname{Aut}(G)rvert$? What if we also assume $G$ is abelian? (I know that $lvert operatorname{Aut}(H)rvert space big| space lvert operatorname{Aut}(G)rvert$ if $G$ is cyclic).
group-theory finite-groups abelian-groups
edited Jan 4 at 3:03
the_fox
2,69221535
$endgroup$
add a comment |
$begingroup$
Let $H$ be a subgroup of a finite group $G$. Is it true that $lvert operatorname{Aut}(H)rvert$ divides $lvert operatorname{Aut}(G)rvert$? What if we also assume $G$ is abelian? (I know that $lvert operatorname{Aut}(H)rvert space big| space lvert operatorname{Aut}(G)rvert$ if $G$ is cyclic).
group-theory finite-groups abelian-groups
edited Jan 4 at 3:03
the_fox
2,69221535
$endgroup$
$begingroup$
This is relevant: mathoverflow.net/questions/9749/…
$endgroup$
– hjhjhj57
Apr 2 '15 at 5:31
add a comment |
$begingroup$
Let $H$ be a subgroup of a finite group $G$. Is it true that $lvert operatorname{Aut}(H)rvert$ divides $lvert operatorname{Aut}(G)rvert$? What if we also assume $G$ is abelian? (I know that $lvert operatorname{Aut}(H)rvert space big| space lvert operatorname{Aut}(G)rvert$ if $G$ is cyclic).
group-theory finite-groups abelian-groups
edited Jan 4 at 3:03
the_fox
2,69221535
$endgroup$
Let $H$ be a subgroup of a finite group $G$. Is it true that $lvert operatorname{Aut}(H)rvert$ divides $lvert operatorname{Aut}(G)rvert$? What if we also assume $G$ is abelian? (I know that $lvert operatorname{Aut}(H)rvert space big| space lvert operatorname{Aut}(G)rvert$ if $G$ is cyclic).
group-theory finite-groups abelian-groups
group-theory finite-groups abelian-groups
edited Jan 4 at 3:03
the_fox
2,69221535
edited Jan 4 at 3:03
the_fox
2,69221535
edited Jan 4 at 3:03
the_fox
2,69221535
edited Jan 4 at 3:03
the_fox
2,69221535
edited Jan 4 at 3:03
the_fox
2,69221535
2,69221535
asked Apr 2 '15 at 4:50
user228168
$begingroup$
This is relevant: mathoverflow.net/questions/9749/…
$endgroup$
– hjhjhj57
Apr 2 '15 at 5:31
add a comment |
$begingroup$
This is relevant: mathoverflow.net/questions/9749/…
$endgroup$
– hjhjhj57
Apr 2 '15 at 5:31
$begingroup$
This is relevant: mathoverflow.net/questions/9749/…
$endgroup$
– hjhjhj57
Apr 2 '15 at 5:31
$begingroup$
This is relevant: mathoverflow.net/questions/9749/…
$endgroup$
– hjhjhj57
Apr 2 '15 at 5:31
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
It's not even true for abelian groups in general. Take $H=C_2times C_2$ as a subgroup of $G=C_4times C_2$. Then $lvert operatorname{Aut}(G)rvert=8$, while $lvert operatorname{Aut}(H)rvert=6$.
edited Jan 4 at 3:06
the_fox
2,69221535
answered Apr 2 '15 at 5:18
verretverret
3,1641922
$endgroup$
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%2f1216853%2ffor-a-subgroup-h-of-a-finite-group-g-when-does-lvert-operatornameaut%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
$begingroup$
It's not even true for abelian groups in general. Take $H=C_2times C_2$ as a subgroup of $G=C_4times C_2$. Then $lvert operatorname{Aut}(G)rvert=8$, while $lvert operatorname{Aut}(H)rvert=6$.
edited Jan 4 at 3:06
the_fox
2,69221535
answered Apr 2 '15 at 5:18
verretverret
3,1641922
$endgroup$
add a comment |
$begingroup$
It's not even true for abelian groups in general. Take $H=C_2times C_2$ as a subgroup of $G=C_4times C_2$. Then $lvert operatorname{Aut}(G)rvert=8$, while $lvert operatorname{Aut}(H)rvert=6$.
edited Jan 4 at 3:06
the_fox
2,69221535
answered Apr 2 '15 at 5:18
verretverret
3,1641922
$endgroup$
add a comment |
$begingroup$
It's not even true for abelian groups in general. Take $H=C_2times C_2$ as a subgroup of $G=C_4times C_2$. Then $lvert operatorname{Aut}(G)rvert=8$, while $lvert operatorname{Aut}(H)rvert=6$.
edited Jan 4 at 3:06
the_fox
2,69221535
answered Apr 2 '15 at 5:18
verretverret
3,1641922
$endgroup$
It's not even true for abelian groups in general. Take $H=C_2times C_2$ as a subgroup of $G=C_4times C_2$. Then $lvert operatorname{Aut}(G)rvert=8$, while $lvert operatorname{Aut}(H)rvert=6$.
edited Jan 4 at 3:06
the_fox
2,69221535
answered Apr 2 '15 at 5:18
verretverret
3,1641922
edited Jan 4 at 3:06
the_fox
2,69221535
edited Jan 4 at 3:06
the_fox
2,69221535
edited Jan 4 at 3:06
the_fox
2,69221535
2,69221535
answered Apr 2 '15 at 5:18
verretverret
3,1641922
answered Apr 2 '15 at 5:18
verretverret
3,1641922
answered Apr 2 '15 at 5:18
verretverret
3,1641922
3,1641922
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%2f1216853%2ffor-a-subgroup-h-of-a-finite-group-g-when-does-lvert-operatornameaut%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
$begingroup$
This is relevant: mathoverflow.net/questions/9749/…
$endgroup$
– hjhjhj57
Apr 2 '15 at 5:31