If $G$ is a cyclic group of order $d$ with generator $a$, show that $dfrac{mathbb{Z}}{dmathbb{Z}}$ is...
$begingroup$
This question is an exact duplicate of:
Let $G = langle arangle$ a cyclic group of order m. Prove that G is isomorphic with $Z_m$. [duplicate]
1 answer
We have to $|G|=d$ and $G=left<aright>={a^n : n in mathbb{Z}}$ by definition of cyclic group.
To show what you are asking, I want to define a function, not necessarily this
$$phi : mathbb{Z} to G text{ such that } z mapsto phi(z)=a^d$$
and use the theorem: Let $varphi:G to G'$ be a group homomorphism, then
$$dfrac{G}{ker(varphi)} simeq Im(varphi)$$
I need your help to define well a function that serves me.
group-theory group-isomorphism group-homomorphism
asked Dec 8 '18 at 19:45
wavilson ferreirawavilson ferreira
517
$endgroup$
marked as duplicate by Dietrich Burde group-theory
Users with the group-theory badge can single-handedly close group-theory questions as duplicates and reopen them as needed.
StackExchange.ready(function() {
if (StackExchange.options.isMobile) return;
$('.dupe-hammer-message-hover:not(.hover-bound)').each(function() {
var $hover = $(this).addClass('hover-bound'),
$msg = $hover.siblings('.dupe-hammer-message');
$hover.hover(
function() {
$hover.showInfoMessage('', {
messageElement: $msg.clone().show(),
transient: false,
position: { my: 'bottom left', at: 'top center', offsetTop: -7 },
dismissable: false,
relativeToBody: true
});
},
function() {
StackExchange.helpers.removeMessages();
}
);
});
});
Dec 8 '18 at 19:56
This question was marked as an exact duplicate of an existing question.
add a comment |
$begingroup$
This question is an exact duplicate of:
Let $G = langle arangle$ a cyclic group of order m. Prove that G is isomorphic with $Z_m$. [duplicate]
1 answer
We have to $|G|=d$ and $G=left<aright>={a^n : n in mathbb{Z}}$ by definition of cyclic group.
To show what you are asking, I want to define a function, not necessarily this
$$phi : mathbb{Z} to G text{ such that } z mapsto phi(z)=a^d$$
and use the theorem: Let $varphi:G to G'$ be a group homomorphism, then
$$dfrac{G}{ker(varphi)} simeq Im(varphi)$$
I need your help to define well a function that serves me.
group-theory group-isomorphism group-homomorphism
asked Dec 8 '18 at 19:45
wavilson ferreirawavilson ferreira
517
$endgroup$
marked as duplicate by Dietrich Burde group-theory
Users with the group-theory badge can single-handedly close group-theory questions as duplicates and reopen them as needed.
StackExchange.ready(function() {
if (StackExchange.options.isMobile) return;
$('.dupe-hammer-message-hover:not(.hover-bound)').each(function() {
var $hover = $(this).addClass('hover-bound'),
$msg = $hover.siblings('.dupe-hammer-message');
$hover.hover(
function() {
$hover.showInfoMessage('', {
messageElement: $msg.clone().show(),
transient: false,
position: { my: 'bottom left', at: 'top center', offsetTop: -7 },
dismissable: false,
relativeToBody: true
});
},
function() {
StackExchange.helpers.removeMessages();
}
);
});
});
Dec 8 '18 at 19:56
This question was marked as an exact duplicate of an existing question.
add a comment |
$begingroup$
This question is an exact duplicate of:
Let $G = langle arangle$ a cyclic group of order m. Prove that G is isomorphic with $Z_m$. [duplicate]
1 answer
We have to $|G|=d$ and $G=left<aright>={a^n : n in mathbb{Z}}$ by definition of cyclic group.
To show what you are asking, I want to define a function, not necessarily this
$$phi : mathbb{Z} to G text{ such that } z mapsto phi(z)=a^d$$
and use the theorem: Let $varphi:G to G'$ be a group homomorphism, then
$$dfrac{G}{ker(varphi)} simeq Im(varphi)$$
I need your help to define well a function that serves me.
group-theory group-isomorphism group-homomorphism
asked Dec 8 '18 at 19:45
wavilson ferreirawavilson ferreira
517
$endgroup$
This question is an exact duplicate of:
Let $G = langle arangle$ a cyclic group of order m. Prove that G is isomorphic with $Z_m$. [duplicate]
1 answer
We have to $|G|=d$ and $G=left<aright>={a^n : n in mathbb{Z}}$ by definition of cyclic group.
To show what you are asking, I want to define a function, not necessarily this
$$phi : mathbb{Z} to G text{ such that } z mapsto phi(z)=a^d$$
and use the theorem: Let $varphi:G to G'$ be a group homomorphism, then
$$dfrac{G}{ker(varphi)} simeq Im(varphi)$$
I need your help to define well a function that serves me.
This question is an exact duplicate of:
Let $G = langle arangle$ a cyclic group of order m. Prove that G is isomorphic with $Z_m$. [duplicate]
1 answer
group-theory group-isomorphism group-homomorphism
group-theory group-isomorphism group-homomorphism
asked Dec 8 '18 at 19:45
wavilson ferreirawavilson ferreira
517
asked Dec 8 '18 at 19:45
wavilson ferreirawavilson ferreira
517
asked Dec 8 '18 at 19:45
wavilson ferreirawavilson ferreira
517
asked Dec 8 '18 at 19:45
wavilson ferreirawavilson ferreira
517
asked Dec 8 '18 at 19:45
wavilson ferreirawavilson ferreira
517
517
marked as duplicate by Dietrich Burde group-theory
Users with the group-theory badge can single-handedly close group-theory questions as duplicates and reopen them as needed.
StackExchange.ready(function() {
if (StackExchange.options.isMobile) return;
$('.dupe-hammer-message-hover:not(.hover-bound)').each(function() {
var $hover = $(this).addClass('hover-bound'),
$msg = $hover.siblings('.dupe-hammer-message');
$hover.hover(
function() {
$hover.showInfoMessage('', {
messageElement: $msg.clone().show(),
transient: false,
position: { my: 'bottom left', at: 'top center', offsetTop: -7 },
dismissable: false,
relativeToBody: true
});
},
function() {
StackExchange.helpers.removeMessages();
}
);
});
});
Dec 8 '18 at 19:56
This question was marked as an exact duplicate of an existing question.
marked as duplicate by Dietrich Burde group-theory
Users with the group-theory badge can single-handedly close group-theory questions as duplicates and reopen them as needed.
StackExchange.ready(function() {
if (StackExchange.options.isMobile) return;
$('.dupe-hammer-message-hover:not(.hover-bound)').each(function() {
var $hover = $(this).addClass('hover-bound'),
$msg = $hover.siblings('.dupe-hammer-message');
$hover.hover(
function() {
$hover.showInfoMessage('', {
messageElement: $msg.clone().show(),
transient: false,
position: { my: 'bottom left', at: 'top center', offsetTop: -7 },
dismissable: false,
relativeToBody: true
});
},
function() {
StackExchange.helpers.removeMessages();
}
);
});
});
Dec 8 '18 at 19:56
This question was marked as an exact duplicate of an existing question.
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Hint: Define $phi (x)=a^x$, for each $xinmathbb Z$.
answered Dec 8 '18 at 20:02
Chris CusterChris Custer
14.2k3827
$endgroup$
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Hint: Define $phi (x)=a^x$, for each $xinmathbb Z$.
answered Dec 8 '18 at 20:02
Chris CusterChris Custer
14.2k3827
$endgroup$
add a comment |
$begingroup$
Hint: Define $phi (x)=a^x$, for each $xinmathbb Z$.
answered Dec 8 '18 at 20:02
Chris CusterChris Custer
14.2k3827
$endgroup$
add a comment |
$begingroup$
Hint: Define $phi (x)=a^x$, for each $xinmathbb Z$.
answered Dec 8 '18 at 20:02
Chris CusterChris Custer
14.2k3827
$endgroup$
Hint: Define $phi (x)=a^x$, for each $xinmathbb Z$.
answered Dec 8 '18 at 20:02
Chris CusterChris Custer
14.2k3827
answered Dec 8 '18 at 20:02
Chris CusterChris Custer
14.2k3827
answered Dec 8 '18 at 20:02
Chris CusterChris Custer
14.2k3827
answered Dec 8 '18 at 20:02
Chris CusterChris Custer
14.2k3827
14.2k3827
add a comment |
add a comment |
