Isomorphism between two groups
$begingroup$
Prove that $langle Bbb Z/nBbb Z, +rangle$ and $langle U_{n}, {}cdot{}rangle$ are isomorphic binary structure where $U_n$ is roots of unity and $Bbb Z/nBbb Z$ is integers modulo $n$.
I know that for isomorphic binary structure, we define a function between groups and we should check homomorphism property and bijection. But I can not define a function. Please help me, if you have any good idea.
abstract-algebra
edited Dec 4 '18 at 9:50
Chinnapparaj R
5,4942928
asked Dec 4 '18 at 9:07
mathsstudentmathsstudent
386
$endgroup$
|
show 1 more comment
$begingroup$
Prove that $langle Bbb Z/nBbb Z, +rangle$ and $langle U_{n}, {}cdot{}rangle$ are isomorphic binary structure where $U_n$ is roots of unity and $Bbb Z/nBbb Z$ is integers modulo $n$.
I know that for isomorphic binary structure, we define a function between groups and we should check homomorphism property and bijection. But I can not define a function. Please help me, if you have any good idea.
abstract-algebra
edited Dec 4 '18 at 9:50
Chinnapparaj R
5,4942928
asked Dec 4 '18 at 9:07
mathsstudentmathsstudent
386
$endgroup$
$begingroup$
That's better. Now, if you're stuck on a general problem (like here, "show that for all $n$, something something"), it usually helps to check a few small examples and see if you can get some insight. What about $Bbb Z/Bbb Z_2$ and $U_2$? What about $Bbb Z/3Bbb Z$ and $U_3$? Can you find isomorhisms there? What about $4$ and $5$? Does that generalize in any way?
$endgroup$
– Arthur
Dec 4 '18 at 9:16
$begingroup$
You are right. Thank you. I can define a map between $Z/3Z$ and $U3$ for example : $f(3k)$ = $(3k)^i$ or can not ?
$endgroup$
– mathsstudent
Dec 4 '18 at 9:20
$begingroup$
What are the elements of $Bbb Z/3Bbb Z$, and what are the elements of $U_3$?
$endgroup$
– Arthur
Dec 4 '18 at 9:22
$begingroup$
$Z/3Z= {0, 3, 6, 9, 12,....}$ and $U3={1,t, t^{2}}$ where $t=e^{2pi(i)/3}$
$endgroup$
– mathsstudent
Dec 4 '18 at 9:27
$begingroup$
No, $0, 3, 6, 9, ldots$ is $3Bbb Z$, not $Bbb Z/3Bbb Z$. The three elements of $Bbb Z/3Bbb Z$ are ${ldots, -3, 0, 3, 6, ldots}$ and ${ldots,-2, 1, 4, 7, ldots}$ and ${ldots,-1, 2, 5, 8, ldots}$, usually called $[0], [1]$ and $[2]$.
$endgroup$
– Arthur
Dec 4 '18 at 9:29
|
show 1 more comment
$begingroup$
Prove that $langle Bbb Z/nBbb Z, +rangle$ and $langle U_{n}, {}cdot{}rangle$ are isomorphic binary structure where $U_n$ is roots of unity and $Bbb Z/nBbb Z$ is integers modulo $n$.
I know that for isomorphic binary structure, we define a function between groups and we should check homomorphism property and bijection. But I can not define a function. Please help me, if you have any good idea.
abstract-algebra
edited Dec 4 '18 at 9:50
Chinnapparaj R
5,4942928
asked Dec 4 '18 at 9:07
mathsstudentmathsstudent
386
$endgroup$
Prove that $langle Bbb Z/nBbb Z, +rangle$ and $langle U_{n}, {}cdot{}rangle$ are isomorphic binary structure where $U_n$ is roots of unity and $Bbb Z/nBbb Z$ is integers modulo $n$.
I know that for isomorphic binary structure, we define a function between groups and we should check homomorphism property and bijection. But I can not define a function. Please help me, if you have any good idea.
abstract-algebra
abstract-algebra
edited Dec 4 '18 at 9:50
Chinnapparaj R
5,4942928
asked Dec 4 '18 at 9:07
mathsstudentmathsstudent
386
edited Dec 4 '18 at 9:50
Chinnapparaj R
5,4942928
asked Dec 4 '18 at 9:07
mathsstudentmathsstudent
386
edited Dec 4 '18 at 9:50
Chinnapparaj R
5,4942928
edited Dec 4 '18 at 9:50
Chinnapparaj R
5,4942928
edited Dec 4 '18 at 9:50
Chinnapparaj R
5,4942928
5,4942928
asked Dec 4 '18 at 9:07
mathsstudentmathsstudent
386
asked Dec 4 '18 at 9:07
mathsstudentmathsstudent
386
asked Dec 4 '18 at 9:07
mathsstudentmathsstudent
386
386
$begingroup$
That's better. Now, if you're stuck on a general problem (like here, "show that for all $n$, something something"), it usually helps to check a few small examples and see if you can get some insight. What about $Bbb Z/Bbb Z_2$ and $U_2$? What about $Bbb Z/3Bbb Z$ and $U_3$? Can you find isomorhisms there? What about $4$ and $5$? Does that generalize in any way?
$endgroup$
– Arthur
Dec 4 '18 at 9:16
$begingroup$
You are right. Thank you. I can define a map between $Z/3Z$ and $U3$ for example : $f(3k)$ = $(3k)^i$ or can not ?
$endgroup$
– mathsstudent
Dec 4 '18 at 9:20
$begingroup$
What are the elements of $Bbb Z/3Bbb Z$, and what are the elements of $U_3$?
$endgroup$
– Arthur
Dec 4 '18 at 9:22
$begingroup$
$Z/3Z= {0, 3, 6, 9, 12,....}$ and $U3={1,t, t^{2}}$ where $t=e^{2pi(i)/3}$
$endgroup$
– mathsstudent
Dec 4 '18 at 9:27
$begingroup$
No, $0, 3, 6, 9, ldots$ is $3Bbb Z$, not $Bbb Z/3Bbb Z$. The three elements of $Bbb Z/3Bbb Z$ are ${ldots, -3, 0, 3, 6, ldots}$ and ${ldots,-2, 1, 4, 7, ldots}$ and ${ldots,-1, 2, 5, 8, ldots}$, usually called $[0], [1]$ and $[2]$.
$endgroup$
– Arthur
Dec 4 '18 at 9:29
|
show 1 more comment
$begingroup$
That's better. Now, if you're stuck on a general problem (like here, "show that for all $n$, something something"), it usually helps to check a few small examples and see if you can get some insight. What about $Bbb Z/Bbb Z_2$ and $U_2$? What about $Bbb Z/3Bbb Z$ and $U_3$? Can you find isomorhisms there? What about $4$ and $5$? Does that generalize in any way?
$endgroup$
– Arthur
Dec 4 '18 at 9:16
$begingroup$
You are right. Thank you. I can define a map between $Z/3Z$ and $U3$ for example : $f(3k)$ = $(3k)^i$ or can not ?
$endgroup$
– mathsstudent
Dec 4 '18 at 9:20
$begingroup$
What are the elements of $Bbb Z/3Bbb Z$, and what are the elements of $U_3$?
$endgroup$
– Arthur
Dec 4 '18 at 9:22
$begingroup$
$Z/3Z= {0, 3, 6, 9, 12,....}$ and $U3={1,t, t^{2}}$ where $t=e^{2pi(i)/3}$
$endgroup$
– mathsstudent
Dec 4 '18 at 9:27
$begingroup$
No, $0, 3, 6, 9, ldots$ is $3Bbb Z$, not $Bbb Z/3Bbb Z$. The three elements of $Bbb Z/3Bbb Z$ are ${ldots, -3, 0, 3, 6, ldots}$ and ${ldots,-2, 1, 4, 7, ldots}$ and ${ldots,-1, 2, 5, 8, ldots}$, usually called $[0], [1]$ and $[2]$.
$endgroup$
– Arthur
Dec 4 '18 at 9:29
$begingroup$
That's better. Now, if you're stuck on a general problem (like here, "show that for all $n$, something something"), it usually helps to check a few small examples and see if you can get some insight. What about $Bbb Z/Bbb Z_2$ and $U_2$? What about $Bbb Z/3Bbb Z$ and $U_3$? Can you find isomorhisms there? What about $4$ and $5$? Does that generalize in any way?
$endgroup$
– Arthur
Dec 4 '18 at 9:16
$begingroup$
That's better. Now, if you're stuck on a general problem (like here, "show that for all $n$, something something"), it usually helps to check a few small examples and see if you can get some insight. What about $Bbb Z/Bbb Z_2$ and $U_2$? What about $Bbb Z/3Bbb Z$ and $U_3$? Can you find isomorhisms there? What about $4$ and $5$? Does that generalize in any way?
$endgroup$
– Arthur
Dec 4 '18 at 9:16
$begingroup$
You are right. Thank you. I can define a map between $Z/3Z$ and $U3$ for example : $f(3k)$ = $(3k)^i$ or can not ?
$endgroup$
– mathsstudent
Dec 4 '18 at 9:20
$begingroup$
You are right. Thank you. I can define a map between $Z/3Z$ and $U3$ for example : $f(3k)$ = $(3k)^i$ or can not ?
$endgroup$
– mathsstudent
Dec 4 '18 at 9:20
$begingroup$
What are the elements of $Bbb Z/3Bbb Z$, and what are the elements of $U_3$?
$endgroup$
– Arthur
Dec 4 '18 at 9:22
$begingroup$
What are the elements of $Bbb Z/3Bbb Z$, and what are the elements of $U_3$?
$endgroup$
– Arthur
Dec 4 '18 at 9:22
$begingroup$
$Z/3Z= {0, 3, 6, 9, 12,....}$ and $U3={1,t, t^{2}}$ where $t=e^{2pi(i)/3}$
$endgroup$
– mathsstudent
Dec 4 '18 at 9:27
$begingroup$
$Z/3Z= {0, 3, 6, 9, 12,....}$ and $U3={1,t, t^{2}}$ where $t=e^{2pi(i)/3}$
$endgroup$
– mathsstudent
Dec 4 '18 at 9:27
$begingroup$
No, $0, 3, 6, 9, ldots$ is $3Bbb Z$, not $Bbb Z/3Bbb Z$. The three elements of $Bbb Z/3Bbb Z$ are ${ldots, -3, 0, 3, 6, ldots}$ and ${ldots,-2, 1, 4, 7, ldots}$ and ${ldots,-1, 2, 5, 8, ldots}$, usually called $[0], [1]$ and $[2]$.
$endgroup$
– Arthur
Dec 4 '18 at 9:29
$begingroup$
No, $0, 3, 6, 9, ldots$ is $3Bbb Z$, not $Bbb Z/3Bbb Z$. The three elements of $Bbb Z/3Bbb Z$ are ${ldots, -3, 0, 3, 6, ldots}$ and ${ldots,-2, 1, 4, 7, ldots}$ and ${ldots,-1, 2, 5, 8, ldots}$, usually called $[0], [1]$ and $[2]$.
$endgroup$
– Arthur
Dec 4 '18 at 9:29
|
show 1 more comment
1 Answer
1
active
oldest
votes
$begingroup$
Outline:
$$U_n={1, zeta,zeta^2,cdots, zeta^{n-1}}=langle zeta rangle$$ where $zeta=e^frac{2 pi i}{n}$
For a sake of simplicity, identify $Bbb Z/ n Bbb Z$ with $Bbb Z_n$. Here $Bbb Z_n =langle 1 rangle$. Then the map $$ f:Bbb Z_n ni 1^i mapsto zeta^i in U_n$$ is an isomorphism (!). Here $1^n$ means $underbrace{1+1+cdots+1}_{n ;times}$
answered Dec 4 '18 at 9:25
Chinnapparaj RChinnapparaj R
5,4942928
$endgroup$
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%2f3025320%2fisomorphism-between-two-groups%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
$begingroup$
Outline:
$$U_n={1, zeta,zeta^2,cdots, zeta^{n-1}}=langle zeta rangle$$ where $zeta=e^frac{2 pi i}{n}$
For a sake of simplicity, identify $Bbb Z/ n Bbb Z$ with $Bbb Z_n$. Here $Bbb Z_n =langle 1 rangle$. Then the map $$ f:Bbb Z_n ni 1^i mapsto zeta^i in U_n$$ is an isomorphism (!). Here $1^n$ means $underbrace{1+1+cdots+1}_{n ;times}$
answered Dec 4 '18 at 9:25
Chinnapparaj RChinnapparaj R
5,4942928
$endgroup$
add a comment |
$begingroup$
Outline:
$$U_n={1, zeta,zeta^2,cdots, zeta^{n-1}}=langle zeta rangle$$ where $zeta=e^frac{2 pi i}{n}$
For a sake of simplicity, identify $Bbb Z/ n Bbb Z$ with $Bbb Z_n$. Here $Bbb Z_n =langle 1 rangle$. Then the map $$ f:Bbb Z_n ni 1^i mapsto zeta^i in U_n$$ is an isomorphism (!). Here $1^n$ means $underbrace{1+1+cdots+1}_{n ;times}$
answered Dec 4 '18 at 9:25
Chinnapparaj RChinnapparaj R
5,4942928
$endgroup$
add a comment |
$begingroup$
Outline:
$$U_n={1, zeta,zeta^2,cdots, zeta^{n-1}}=langle zeta rangle$$ where $zeta=e^frac{2 pi i}{n}$
For a sake of simplicity, identify $Bbb Z/ n Bbb Z$ with $Bbb Z_n$. Here $Bbb Z_n =langle 1 rangle$. Then the map $$ f:Bbb Z_n ni 1^i mapsto zeta^i in U_n$$ is an isomorphism (!). Here $1^n$ means $underbrace{1+1+cdots+1}_{n ;times}$
answered Dec 4 '18 at 9:25
Chinnapparaj RChinnapparaj R
5,4942928
$endgroup$
Outline:
$$U_n={1, zeta,zeta^2,cdots, zeta^{n-1}}=langle zeta rangle$$ where $zeta=e^frac{2 pi i}{n}$
For a sake of simplicity, identify $Bbb Z/ n Bbb Z$ with $Bbb Z_n$. Here $Bbb Z_n =langle 1 rangle$. Then the map $$ f:Bbb Z_n ni 1^i mapsto zeta^i in U_n$$ is an isomorphism (!). Here $1^n$ means $underbrace{1+1+cdots+1}_{n ;times}$
answered Dec 4 '18 at 9:25
Chinnapparaj RChinnapparaj R
5,4942928
answered Dec 4 '18 at 9:25
Chinnapparaj RChinnapparaj R
5,4942928
answered Dec 4 '18 at 9:25
Chinnapparaj RChinnapparaj R
5,4942928
answered Dec 4 '18 at 9:25
Chinnapparaj RChinnapparaj R
5,4942928
5,4942928
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%2f3025320%2fisomorphism-between-two-groups%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
$begingroup$
That's better. Now, if you're stuck on a general problem (like here, "show that for all $n$, something something"), it usually helps to check a few small examples and see if you can get some insight. What about $Bbb Z/Bbb Z_2$ and $U_2$? What about $Bbb Z/3Bbb Z$ and $U_3$? Can you find isomorhisms there? What about $4$ and $5$? Does that generalize in any way?
$endgroup$
– Arthur
Dec 4 '18 at 9:16
$begingroup$
You are right. Thank you. I can define a map between $Z/3Z$ and $U3$ for example : $f(3k)$ = $(3k)^i$ or can not ?
$endgroup$
– mathsstudent
Dec 4 '18 at 9:20
$begingroup$
What are the elements of $Bbb Z/3Bbb Z$, and what are the elements of $U_3$?
$endgroup$
– Arthur
Dec 4 '18 at 9:22
$begingroup$
$Z/3Z= {0, 3, 6, 9, 12,....}$ and $U3={1,t, t^{2}}$ where $t=e^{2pi(i)/3}$
$endgroup$
– mathsstudent
Dec 4 '18 at 9:27
$begingroup$
No, $0, 3, 6, 9, ldots$ is $3Bbb Z$, not $Bbb Z/3Bbb Z$. The three elements of $Bbb Z/3Bbb Z$ are ${ldots, -3, 0, 3, 6, ldots}$ and ${ldots,-2, 1, 4, 7, ldots}$ and ${ldots,-1, 2, 5, 8, ldots}$, usually called $[0], [1]$ and $[2]$.
$endgroup$
– Arthur
Dec 4 '18 at 9:29