Mixing
Homomorphism from $S_n$ to $D_k$
$begingroup$
Let $k$ and $n$ be natural numbers so that $k>2$, $n>4$ and $k$ is not divisible by $3$. How many homomorphisms from the symmetric group $S_n$ to the dihedral group $D_k$ are there?
abstract-algebra finite-groups
edited Jan 12 at 21:23
Bernard
121k740116
asked Jan 12 at 21:09
ig97ig97
155
$endgroup$
add a comment |
$begingroup$
Let $k$ and $n$ be natural numbers so that $k>2$, $n>4$ and $k$ is not divisible by $3$. How many homomorphisms from the symmetric group $S_n$ to the dihedral group $D_k$ are there?
abstract-algebra finite-groups
edited Jan 12 at 21:23
Bernard
121k740116
asked Jan 12 at 21:09
ig97ig97
155
$endgroup$
1
$begingroup$
Please edit the question to show us what you tried. You can begin by working out some small examples by hand.
$endgroup$
– Ethan Bolker
Jan 12 at 21:13
add a comment |
$begingroup$
Let $k$ and $n$ be natural numbers so that $k>2$, $n>4$ and $k$ is not divisible by $3$. How many homomorphisms from the symmetric group $S_n$ to the dihedral group $D_k$ are there?
abstract-algebra finite-groups
edited Jan 12 at 21:23
Bernard
121k740116
asked Jan 12 at 21:09
ig97ig97
155
$endgroup$
Let $k$ and $n$ be natural numbers so that $k>2$, $n>4$ and $k$ is not divisible by $3$. How many homomorphisms from the symmetric group $S_n$ to the dihedral group $D_k$ are there?
abstract-algebra finite-groups
abstract-algebra finite-groups
edited Jan 12 at 21:23
Bernard
121k740116
asked Jan 12 at 21:09
ig97ig97
155
edited Jan 12 at 21:23
Bernard
121k740116
asked Jan 12 at 21:09
ig97ig97
155
edited Jan 12 at 21:23
Bernard
121k740116
edited Jan 12 at 21:23
Bernard
121k740116
edited Jan 12 at 21:23
Bernard
121k740116
121k740116
asked Jan 12 at 21:09
ig97ig97
155
asked Jan 12 at 21:09
ig97ig97
155
asked Jan 12 at 21:09
ig97ig97
155
155
1
$begingroup$
Please edit the question to show us what you tried. You can begin by working out some small examples by hand.
$endgroup$
– Ethan Bolker
Jan 12 at 21:13
add a comment |
1
$begingroup$
Please edit the question to show us what you tried. You can begin by working out some small examples by hand.
$endgroup$
– Ethan Bolker
Jan 12 at 21:13
1
1
$begingroup$
Please edit the question to show us what you tried. You can begin by working out some small examples by hand.
$endgroup$
– Ethan Bolker
Jan 12 at 21:13
$begingroup$
Please edit the question to show us what you tried. You can begin by working out some small examples by hand.
$endgroup$
– Ethan Bolker
Jan 12 at 21:13
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
There are only three normal subgroups of $S_n$. The resulting quotients have orders $1$, $2$, and $n! $. The third option is not possible because $k$ is not divisible by $3$. So the answer is one more than the number of elements of $D_k$ of order $2$.
answered Jan 13 at 0:21
Matt SamuelMatt Samuel
38.4k63768
$endgroup$
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%2f3071397%2fhomomorphism-from-s-n-to-d-k%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
$begingroup$
There are only three normal subgroups of $S_n$. The resulting quotients have orders $1$, $2$, and $n! $. The third option is not possible because $k$ is not divisible by $3$. So the answer is one more than the number of elements of $D_k$ of order $2$.
answered Jan 13 at 0:21
Matt SamuelMatt Samuel
38.4k63768
$endgroup$
add a comment |
$begingroup$
There are only three normal subgroups of $S_n$. The resulting quotients have orders $1$, $2$, and $n! $. The third option is not possible because $k$ is not divisible by $3$. So the answer is one more than the number of elements of $D_k$ of order $2$.
answered Jan 13 at 0:21
Matt SamuelMatt Samuel
38.4k63768
$endgroup$
add a comment |
$begingroup$
There are only three normal subgroups of $S_n$. The resulting quotients have orders $1$, $2$, and $n! $. The third option is not possible because $k$ is not divisible by $3$. So the answer is one more than the number of elements of $D_k$ of order $2$.
answered Jan 13 at 0:21
Matt SamuelMatt Samuel
38.4k63768
$endgroup$
There are only three normal subgroups of $S_n$. The resulting quotients have orders $1$, $2$, and $n! $. The third option is not possible because $k$ is not divisible by $3$. So the answer is one more than the number of elements of $D_k$ of order $2$.
answered Jan 13 at 0:21
Matt SamuelMatt Samuel
38.4k63768
answered Jan 13 at 0:21
Matt SamuelMatt Samuel
38.4k63768
answered Jan 13 at 0:21
Matt SamuelMatt Samuel
38.4k63768
answered Jan 13 at 0:21
Matt SamuelMatt Samuel
38.4k63768
38.4k63768
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%2f3071397%2fhomomorphism-from-s-n-to-d-k%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
1
$begingroup$
Please edit the question to show us what you tried. You can begin by working out some small examples by hand.
$endgroup$
– Ethan Bolker
Jan 12 at 21:13