$X$ has a normal group $Y$ so $[X: Y]=4$.
$begingroup$
Consider the following set: $$X = {sigma in S_6 | sigma(k) is odd if and only if k is odd}.$$
I would like to show that $X$ has a normal group $Y$ so $[X: Y]=4$.
How should I prove this theorem?
abstract-algebra group-theory
edited Jan 12 at 18:46
Shaun
10.5k113687
asked Jan 12 at 17:53
BadukBaduk
493
$endgroup$
add a comment |
$begingroup$
Consider the following set: $$X = {sigma in S_6 | sigma(k) is odd if and only if k is odd}.$$
I would like to show that $X$ has a normal group $Y$ so $[X: Y]=4$.
How should I prove this theorem?
abstract-algebra group-theory
edited Jan 12 at 18:46
Shaun
10.5k113687
asked Jan 12 at 17:53
BadukBaduk
493
$endgroup$
$begingroup$
You should verify that it is a subgroup and that it is normal. Finally, you should figure out how big it is, so you can calculate its index.
$endgroup$
– Cheerful Parsnip
Jan 12 at 18:56
add a comment |
$begingroup$
Consider the following set: $$X = {sigma in S_6 | sigma(k) is odd if and only if k is odd}.$$
I would like to show that $X$ has a normal group $Y$ so $[X: Y]=4$.
How should I prove this theorem?
abstract-algebra group-theory
edited Jan 12 at 18:46
Shaun
10.5k113687
asked Jan 12 at 17:53
BadukBaduk
493
$endgroup$
Consider the following set: $$X = {sigma in S_6 | sigma(k) is odd if and only if k is odd}.$$
I would like to show that $X$ has a normal group $Y$ so $[X: Y]=4$.
How should I prove this theorem?
abstract-algebra group-theory
abstract-algebra group-theory
edited Jan 12 at 18:46
Shaun
10.5k113687
asked Jan 12 at 17:53
BadukBaduk
493
edited Jan 12 at 18:46
Shaun
10.5k113687
asked Jan 12 at 17:53
BadukBaduk
493
edited Jan 12 at 18:46
Shaun
10.5k113687
edited Jan 12 at 18:46
Shaun
10.5k113687
edited Jan 12 at 18:46
Shaun
10.5k113687
10.5k113687
asked Jan 12 at 17:53
BadukBaduk
493
asked Jan 12 at 17:53
BadukBaduk
493
asked Jan 12 at 17:53
BadukBaduk
493
493
$begingroup$
You should verify that it is a subgroup and that it is normal. Finally, you should figure out how big it is, so you can calculate its index.
$endgroup$
– Cheerful Parsnip
Jan 12 at 18:56
add a comment |
$begingroup$
You should verify that it is a subgroup and that it is normal. Finally, you should figure out how big it is, so you can calculate its index.
$endgroup$
– Cheerful Parsnip
Jan 12 at 18:56
$begingroup$
You should verify that it is a subgroup and that it is normal. Finally, you should figure out how big it is, so you can calculate its index.
$endgroup$
– Cheerful Parsnip
Jan 12 at 18:56
$begingroup$
You should verify that it is a subgroup and that it is normal. Finally, you should figure out how big it is, so you can calculate its index.
$endgroup$
– Cheerful Parsnip
Jan 12 at 18:56
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Solution (sketch):
(a) How can we relate X to some group we are familiar with? Prove that $Xcong S_3times S_3$.
Let $S={1,2,3,4,5,6}$ be the set of elements in $S_6$ and let $S_1$ and $S_2$ denote the set of all even resp. odd numbers in $S$.
Let $f:Xrightarrow S_3times S_3$, $sigmamapsto (sigma_1,sigma_2)$, where $sigma_1=sigma|_{S_1}$, $sigma_2=sigma|_{S_2}$. You can show that $f$ is indeed an isomorphism.
(b) Now, we can calculate the size of Y: $$[X:Y]=frac{|X|}{|Y|}=frac{|S_3times S_3|}{Y}=frac{6cdot 6}{|Y|}=4.$$
(c) Take a look at the subgroups of $S_3$ (or, $S_3times S_3$). Which subgroup would be applicable? According to the required size, $Y$ should be isomorphic to $A_3times A_3$.
answered Jan 12 at 20:28
fiatluxfiatlux
132
$endgroup$
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%2f3071192%2fx-has-a-normal-group-y-so-x-y-4%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
$begingroup$
Solution (sketch):
(a) How can we relate X to some group we are familiar with? Prove that $Xcong S_3times S_3$.
Let $S={1,2,3,4,5,6}$ be the set of elements in $S_6$ and let $S_1$ and $S_2$ denote the set of all even resp. odd numbers in $S$.
Let $f:Xrightarrow S_3times S_3$, $sigmamapsto (sigma_1,sigma_2)$, where $sigma_1=sigma|_{S_1}$, $sigma_2=sigma|_{S_2}$. You can show that $f$ is indeed an isomorphism.
(b) Now, we can calculate the size of Y: $$[X:Y]=frac{|X|}{|Y|}=frac{|S_3times S_3|}{Y}=frac{6cdot 6}{|Y|}=4.$$
(c) Take a look at the subgroups of $S_3$ (or, $S_3times S_3$). Which subgroup would be applicable? According to the required size, $Y$ should be isomorphic to $A_3times A_3$.
answered Jan 12 at 20:28
fiatluxfiatlux
132
$endgroup$
add a comment |
$begingroup$
Solution (sketch):
(a) How can we relate X to some group we are familiar with? Prove that $Xcong S_3times S_3$.
Let $S={1,2,3,4,5,6}$ be the set of elements in $S_6$ and let $S_1$ and $S_2$ denote the set of all even resp. odd numbers in $S$.
Let $f:Xrightarrow S_3times S_3$, $sigmamapsto (sigma_1,sigma_2)$, where $sigma_1=sigma|_{S_1}$, $sigma_2=sigma|_{S_2}$. You can show that $f$ is indeed an isomorphism.
(b) Now, we can calculate the size of Y: $$[X:Y]=frac{|X|}{|Y|}=frac{|S_3times S_3|}{Y}=frac{6cdot 6}{|Y|}=4.$$
(c) Take a look at the subgroups of $S_3$ (or, $S_3times S_3$). Which subgroup would be applicable? According to the required size, $Y$ should be isomorphic to $A_3times A_3$.
answered Jan 12 at 20:28
fiatluxfiatlux
132
$endgroup$
add a comment |
$begingroup$
Solution (sketch):
(a) How can we relate X to some group we are familiar with? Prove that $Xcong S_3times S_3$.
Let $S={1,2,3,4,5,6}$ be the set of elements in $S_6$ and let $S_1$ and $S_2$ denote the set of all even resp. odd numbers in $S$.
Let $f:Xrightarrow S_3times S_3$, $sigmamapsto (sigma_1,sigma_2)$, where $sigma_1=sigma|_{S_1}$, $sigma_2=sigma|_{S_2}$. You can show that $f$ is indeed an isomorphism.
(b) Now, we can calculate the size of Y: $$[X:Y]=frac{|X|}{|Y|}=frac{|S_3times S_3|}{Y}=frac{6cdot 6}{|Y|}=4.$$
(c) Take a look at the subgroups of $S_3$ (or, $S_3times S_3$). Which subgroup would be applicable? According to the required size, $Y$ should be isomorphic to $A_3times A_3$.
answered Jan 12 at 20:28
fiatluxfiatlux
132
$endgroup$
Solution (sketch):
(a) How can we relate X to some group we are familiar with? Prove that $Xcong S_3times S_3$.
Let $S={1,2,3,4,5,6}$ be the set of elements in $S_6$ and let $S_1$ and $S_2$ denote the set of all even resp. odd numbers in $S$.
Let $f:Xrightarrow S_3times S_3$, $sigmamapsto (sigma_1,sigma_2)$, where $sigma_1=sigma|_{S_1}$, $sigma_2=sigma|_{S_2}$. You can show that $f$ is indeed an isomorphism.
(b) Now, we can calculate the size of Y: $$[X:Y]=frac{|X|}{|Y|}=frac{|S_3times S_3|}{Y}=frac{6cdot 6}{|Y|}=4.$$
(c) Take a look at the subgroups of $S_3$ (or, $S_3times S_3$). Which subgroup would be applicable? According to the required size, $Y$ should be isomorphic to $A_3times A_3$.
answered Jan 12 at 20:28
fiatluxfiatlux
132
answered Jan 12 at 20:28
fiatluxfiatlux
132
answered Jan 12 at 20:28
fiatluxfiatlux
132
answered Jan 12 at 20:28
fiatluxfiatlux
132
132
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%2f3071192%2fx-has-a-normal-group-y-so-x-y-4%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
$begingroup$
You should verify that it is a subgroup and that it is normal. Finally, you should figure out how big it is, so you can calculate its index.
$endgroup$
– Cheerful Parsnip
Jan 12 at 18:56