What does $mathbbC / mathbbZ$ mean? Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)What exactly does conjugation mean?What does “order” mean in group theory?What does this notation mean: $displaystylelim_leftarrow ,mathbbZ/nmathbbZ$?What does it mean to “Decide to which group $G$ is isomorphic” for a given group $G$?homomorphism $f: mathbbC^* rightarrow mathbbR^*$ with multiplicative groups, prove that kernel of $f$ is infinite.What does “well defined up to isomorphism” mean?Show that $(mathbb Z[x],+)$ and $(mathbb Q_>0,cdot)$ are isomorphic groupsGroup of units in the rings $mathbb I_9 $ and $mathbb I_15$?“multiplicative inverse in the modulo of the larger number” what does that mean?CG-modules: what does this notation mean?
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What does $mathbbC / mathbbZ$ mean?
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)What exactly does conjugation mean?What does “order” mean in group theory?What does this notation mean: $displaystylelim_leftarrow ,mathbbZ/nmathbbZ$?What does it mean to “Decide to which group $G$ is isomorphic” for a given group $G$?homomorphism $f: mathbbC^* rightarrow mathbbR^*$ with multiplicative groups, prove that kernel of $f$ is infinite.What does “well defined up to isomorphism” mean?Show that $(mathbb Z[x],+)$ and $(mathbb Q_>0,cdot)$ are isomorphic groupsGroup of units in the rings $mathbb I_9 $ and $mathbb I_15$?“multiplicative inverse in the modulo of the larger number” what does that mean?CG-modules: what does this notation mean?
$begingroup$
In the following question I need to use $mathbbC / mathbbZ$ :
Prove that $mathbbC / mathbbZ$ is isomorphic to the multiplicative group $(mathbbC^*,cdot, 1)$.
But what does $mathbbC / mathbbZ$ mean? I understand things like $mathbbZ/NmathbbZ$, but cannot grasp $mathbbC/mathbbZ$ . Thanks!
abstract-algebra group-theory modular-arithmetic
asked Mar 31 at 17:42
R4000R4000
191
$endgroup$
add a comment |
$begingroup$
In the following question I need to use $mathbbC / mathbbZ$ :
Prove that $mathbbC / mathbbZ$ is isomorphic to the multiplicative group $(mathbbC^*,cdot, 1)$.
But what does $mathbbC / mathbbZ$ mean? I understand things like $mathbbZ/NmathbbZ$, but cannot grasp $mathbbC/mathbbZ$ . Thanks!
abstract-algebra group-theory modular-arithmetic
asked Mar 31 at 17:42
R4000R4000
191
$endgroup$
1
$begingroup$
An intermediate example might be $mathbbR/mathbbZ$, which has a snappy geometric meaning. Then you can think of $mathbbC/mathbbZ$ as "that, but $times$ a line (= the imaginary axis)" - and again you'll see a nice "shape" being described.
$endgroup$
– Noah Schweber
Mar 31 at 17:53
$begingroup$
What's the difference between $mathbbC/mathbbZ$ and $mathbbZ/NmathbbZ$ that you can't grasp?
$endgroup$
– anomaly
Mar 31 at 18:02
$begingroup$
$mathbbC^*$ does not contain purely integers
$endgroup$
– R4000
Apr 1 at 9:27
add a comment |
$begingroup$
In the following question I need to use $mathbbC / mathbbZ$ :
Prove that $mathbbC / mathbbZ$ is isomorphic to the multiplicative group $(mathbbC^*,cdot, 1)$.
But what does $mathbbC / mathbbZ$ mean? I understand things like $mathbbZ/NmathbbZ$, but cannot grasp $mathbbC/mathbbZ$ . Thanks!
abstract-algebra group-theory modular-arithmetic
asked Mar 31 at 17:42
R4000R4000
191
$endgroup$
In the following question I need to use $mathbbC / mathbbZ$ :
Prove that $mathbbC / mathbbZ$ is isomorphic to the multiplicative group $(mathbbC^*,cdot, 1)$.
But what does $mathbbC / mathbbZ$ mean? I understand things like $mathbbZ/NmathbbZ$, but cannot grasp $mathbbC/mathbbZ$ . Thanks!
abstract-algebra group-theory modular-arithmetic
abstract-algebra group-theory modular-arithmetic
asked Mar 31 at 17:42
R4000R4000
191
asked Mar 31 at 17:42
R4000R4000
191
asked Mar 31 at 17:42
R4000R4000
191
asked Mar 31 at 17:42
R4000R4000
191
asked Mar 31 at 17:42
R4000R4000
191
191
1
$begingroup$
An intermediate example might be $mathbbR/mathbbZ$, which has a snappy geometric meaning. Then you can think of $mathbbC/mathbbZ$ as "that, but $times$ a line (= the imaginary axis)" - and again you'll see a nice "shape" being described.
$endgroup$
– Noah Schweber
Mar 31 at 17:53
$begingroup$
What's the difference between $mathbbC/mathbbZ$ and $mathbbZ/NmathbbZ$ that you can't grasp?
$endgroup$
– anomaly
Mar 31 at 18:02
$begingroup$
$mathbbC^*$ does not contain purely integers
$endgroup$
– R4000
Apr 1 at 9:27
add a comment |
1
$begingroup$
An intermediate example might be $mathbbR/mathbbZ$, which has a snappy geometric meaning. Then you can think of $mathbbC/mathbbZ$ as "that, but $times$ a line (= the imaginary axis)" - and again you'll see a nice "shape" being described.
$endgroup$
– Noah Schweber
Mar 31 at 17:53
$begingroup$
What's the difference between $mathbbC/mathbbZ$ and $mathbbZ/NmathbbZ$ that you can't grasp?
$endgroup$
– anomaly
Mar 31 at 18:02
$begingroup$
$mathbbC^*$ does not contain purely integers
$endgroup$
– R4000
Apr 1 at 9:27
1
1
$begingroup$
An intermediate example might be $mathbbR/mathbbZ$, which has a snappy geometric meaning. Then you can think of $mathbbC/mathbbZ$ as "that, but $times$ a line (= the imaginary axis)" - and again you'll see a nice "shape" being described.
$endgroup$
– Noah Schweber
Mar 31 at 17:53
$begingroup$
An intermediate example might be $mathbbR/mathbbZ$, which has a snappy geometric meaning. Then you can think of $mathbbC/mathbbZ$ as "that, but $times$ a line (= the imaginary axis)" - and again you'll see a nice "shape" being described.
$endgroup$
– Noah Schweber
Mar 31 at 17:53
$begingroup$
What's the difference between $mathbbC/mathbbZ$ and $mathbbZ/NmathbbZ$ that you can't grasp?
$endgroup$
– anomaly
Mar 31 at 18:02
$begingroup$
What's the difference between $mathbbC/mathbbZ$ and $mathbbZ/NmathbbZ$ that you can't grasp?
$endgroup$
– anomaly
Mar 31 at 18:02
$begingroup$
$mathbbC^*$ does not contain purely integers
$endgroup$
– R4000
Apr 1 at 9:27
$begingroup$
$mathbbC^*$ does not contain purely integers
$endgroup$
– R4000
Apr 1 at 9:27
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
$mathbbZ$ is a subgroup of the additive group of $mathbbC$, so you can take the quotient $mathbbC/mathbbZ$.
answered Mar 31 at 17:48
Eclipse SunEclipse Sun
8,1151438
$endgroup$
add a comment |
$begingroup$
$mathbb C/mathbb Z$ is a cylinder.
answered Mar 31 at 17:53
user657324user657324
60110
$endgroup$
$begingroup$
$mathbbC$ under addition has no inherent topological or geometric structure, so this is not really inherent to the construction. But it's probably a good way to visualize it.
$endgroup$
– 6005
Mar 31 at 17:57
1
$begingroup$
@6005: I think the same remark can hold to $mathbb R/mathbb Z$... :)
$endgroup$
– user657324
Mar 31 at 18:02
$begingroup$
Indeed, $mathbbR / mathbbZ$ has no inherent topological or geometric structure :)
$endgroup$
– 6005
Mar 31 at 18:04
add a comment |
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2 Answers
2
active
oldest
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2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
$mathbbZ$ is a subgroup of the additive group of $mathbbC$, so you can take the quotient $mathbbC/mathbbZ$.
answered Mar 31 at 17:48
Eclipse SunEclipse Sun
8,1151438
$endgroup$
add a comment |
$begingroup$
$mathbbZ$ is a subgroup of the additive group of $mathbbC$, so you can take the quotient $mathbbC/mathbbZ$.
answered Mar 31 at 17:48
Eclipse SunEclipse Sun
8,1151438
$endgroup$
add a comment |
$begingroup$
$mathbbZ$ is a subgroup of the additive group of $mathbbC$, so you can take the quotient $mathbbC/mathbbZ$.
answered Mar 31 at 17:48
Eclipse SunEclipse Sun
8,1151438
$endgroup$
$mathbbZ$ is a subgroup of the additive group of $mathbbC$, so you can take the quotient $mathbbC/mathbbZ$.
answered Mar 31 at 17:48
Eclipse SunEclipse Sun
8,1151438
answered Mar 31 at 17:48
Eclipse SunEclipse Sun
8,1151438
answered Mar 31 at 17:48
Eclipse SunEclipse Sun
8,1151438
answered Mar 31 at 17:48
Eclipse SunEclipse Sun
8,1151438
8,1151438
add a comment |
add a comment |
$begingroup$
$mathbb C/mathbb Z$ is a cylinder.
answered Mar 31 at 17:53
user657324user657324
60110
$endgroup$
$begingroup$
$mathbbC$ under addition has no inherent topological or geometric structure, so this is not really inherent to the construction. But it's probably a good way to visualize it.
$endgroup$
– 6005
Mar 31 at 17:57
1
$begingroup$
@6005: I think the same remark can hold to $mathbb R/mathbb Z$... :)
$endgroup$
– user657324
Mar 31 at 18:02
$begingroup$
Indeed, $mathbbR / mathbbZ$ has no inherent topological or geometric structure :)
$endgroup$
– 6005
Mar 31 at 18:04
add a comment |
$begingroup$
$mathbb C/mathbb Z$ is a cylinder.
answered Mar 31 at 17:53
user657324user657324
60110
$endgroup$
$begingroup$
$mathbbC$ under addition has no inherent topological or geometric structure, so this is not really inherent to the construction. But it's probably a good way to visualize it.
$endgroup$
– 6005
Mar 31 at 17:57
1
$begingroup$
@6005: I think the same remark can hold to $mathbb R/mathbb Z$... :)
$endgroup$
– user657324
Mar 31 at 18:02
$begingroup$
Indeed, $mathbbR / mathbbZ$ has no inherent topological or geometric structure :)
$endgroup$
– 6005
Mar 31 at 18:04
add a comment |
$begingroup$
$mathbb C/mathbb Z$ is a cylinder.
answered Mar 31 at 17:53
user657324user657324
60110
$endgroup$
$mathbb C/mathbb Z$ is a cylinder.
answered Mar 31 at 17:53
user657324user657324
60110
answered Mar 31 at 17:53
user657324user657324
60110
answered Mar 31 at 17:53
user657324user657324
60110
answered Mar 31 at 17:53
user657324user657324
60110
60110
$begingroup$
$mathbbC$ under addition has no inherent topological or geometric structure, so this is not really inherent to the construction. But it's probably a good way to visualize it.
$endgroup$
– 6005
Mar 31 at 17:57
1
$begingroup$
@6005: I think the same remark can hold to $mathbb R/mathbb Z$... :)
$endgroup$
– user657324
Mar 31 at 18:02
$begingroup$
Indeed, $mathbbR / mathbbZ$ has no inherent topological or geometric structure :)
$endgroup$
– 6005
Mar 31 at 18:04
add a comment |
$begingroup$
$mathbbC$ under addition has no inherent topological or geometric structure, so this is not really inherent to the construction. But it's probably a good way to visualize it.
$endgroup$
– 6005
Mar 31 at 17:57
1
$begingroup$
@6005: I think the same remark can hold to $mathbb R/mathbb Z$... :)
$endgroup$
– user657324
Mar 31 at 18:02
$begingroup$
Indeed, $mathbbR / mathbbZ$ has no inherent topological or geometric structure :)
$endgroup$
– 6005
Mar 31 at 18:04
$begingroup$
$mathbbC$ under addition has no inherent topological or geometric structure, so this is not really inherent to the construction. But it's probably a good way to visualize it.
$endgroup$
– 6005
Mar 31 at 17:57
$begingroup$
$mathbbC$ under addition has no inherent topological or geometric structure, so this is not really inherent to the construction. But it's probably a good way to visualize it.
$endgroup$
– 6005
Mar 31 at 17:57
1
1
$begingroup$
@6005: I think the same remark can hold to $mathbb R/mathbb Z$... :)
$endgroup$
– user657324
Mar 31 at 18:02
$begingroup$
@6005: I think the same remark can hold to $mathbb R/mathbb Z$... :)
$endgroup$
– user657324
Mar 31 at 18:02
$begingroup$
Indeed, $mathbbR / mathbbZ$ has no inherent topological or geometric structure :)
$endgroup$
– 6005
Mar 31 at 18:04
$begingroup$
Indeed, $mathbbR / mathbbZ$ has no inherent topological or geometric structure :)
$endgroup$
– 6005
Mar 31 at 18:04
add a comment |
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1
$begingroup$
An intermediate example might be $mathbbR/mathbbZ$, which has a snappy geometric meaning. Then you can think of $mathbbC/mathbbZ$ as "that, but $times$ a line (= the imaginary axis)" - and again you'll see a nice "shape" being described.
$endgroup$
– Noah Schweber
Mar 31 at 17:53
$begingroup$
What's the difference between $mathbbC/mathbbZ$ and $mathbbZ/NmathbbZ$ that you can't grasp?
$endgroup$
– anomaly
Mar 31 at 18:02
$begingroup$
$mathbbC^*$ does not contain purely integers
$endgroup$
– R4000
Apr 1 at 9:27