# Isomorphism concepts,( example periods elliptic functions )

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• binbagsss
In summary, the conversation discusses the concept of isomorphism between two sets and the conditions for it to exist. In the given cases, the sets are discrete subgroups of the complex plane and the maps between them are defined by taking integers or integers multiplied together. The relevant operation is addition and the condition for isomorphism is that the ratio of the elements in the second set must not be a rational number.
binbagsss
Hi,
I have the following:

Let ##\Omega ## be a discrete subgroup of ##C##, the complex plane.
If:
i) ##\Omega = \{nw_1 | n \in Z\} ##, then ##\Omega ## is isomorphic to ##Z##.
ii) ##\Omega = \{nw_1 + mw_2 | m,n \in Z\} ## where ##w_1/w_2 \notin R ## , then ##\Omega## is isomorphic to ##Z## x ##Z##

So from what I understand isomorphic is a map that is one to one between two sets that preserves the binary relatione exisising between elements, that is ##f(x*y)=f(x)*f(y)## (1), where ##*## is the operation the map is isomorphic to. So to define a isomorphism you need to define:

- two sets
- the map between them
- the relevant operation which is preserved, defined by (1)

QUESTION 1)
So, my book doens't say which operation, is it addition, it also doesn't say which map - is the map to take the integer with the map ##f = n ## in case i) and ##f=n+m## in case 2, under the operation addition it is then easy to show that (1) is obeyed in both cases?

QUESTION 2)
By the wording it seems to imply the fact that ##w_1/w_2 \notin R ## is significant for there to be an isomorphism to ##Z## x## Z##, I don't at all understand why, can someone explain?

Many thanks in advance

binbagsss said:
QUESTION 1)
So, my book doens't say which operation, is it addition, it also doesn't say which map - is the map to take the integer with the map ##f = n ## in case i) and ##f=n+m## in case 2, under the operation addition it is then easy to show that (1) is obeyed in both cases?

Your second case is not a map from ##\Omega## to ##Z\times Z## and as such cannot be an isomorphism.

binbagsss said:
By the wording it seems to imply the fact that w1/w2∉Rw1/w2∉Rw_1/w_2 \notin R is significant for there to be an isomorphism to ZZZ xZZ Z, I don't at all understand why, can someone explain?
What happens if you take, say, ##w_2 = 2 w_1##?

Orodruin said:
Your second case is not a map from ##\Omega## to ##Z\times Z## and as such cannot be an isomorphism.What happens if you take, say, ##w_2 = 2 w_1##?

So the first case is wrong to?

What would a map to ##Z## x ##Z## look like? two integers multiplied together? do you take ##mn## instead? so the relevant operation is multiplication, not addition?

I don't think I am in a position to answer your second question until I can answer the first..

binbagsss said:
So the first case is wrong to?

What would a map to ##Z## x ##Z## look like? two integers multiplied together?

I don't think I am in a position to answer your second question until I can answer the first..

##Z \times Z## is the set of 2-tuples of integers, not the set of integers.

Orodruin said:
##Z \times Z## is the set of 2-tuples of integers, not the set of integers.
edit: if you take ##w1=2w2## there is ambiguity about what ##n## and ##m## are i.e- the map would no longer be one-to-one?

the map is not ##nm## ? no?

binbagsss said:
edit: if you take ##w1=2w2## there is ambiguity about what ##n## and ##m## are i.e- the map would no longer be one-to-one?

Right. Although, if I am not wrong (I may be, I have been up for quite some time), it would be sufficient to have ##w_1/w_2 \notin \mathbb Q##. Of course, since ##\mathbb Q \subset \mathbb R##, ##w_1/w_2 \notin \mathbb R## implies ##w_1/w_2 \notin \mathbb Q##.

binbagsss said:
the map is not ##nm## ? no?
No, assuming that by ##nm## you mean to multiply ##n## and ##m#, it is an integer, not a 2-tuple of integers.

binbagsss
Orodruin said:
Right. Although, if I am not wrong (I may be, I have been up for quite some time), it would be sufficient to have ##w_1/w_2 \notin \mathbb Q##. Of course, since ##\mathbb Q \subset \mathbb R##, ##w_1/w_2 \notin \mathbb R## implies ##w_1/w_2 \notin \mathbb Q##.No, assuming that by ##nm## you mean to multiply ##n## and ##m#, it is an integer, not a 2-tuple of integers.
Orodruin said:
Right. Although, if I am not wrong (I may be, I have been up for quite some time), it would be sufficient to have ##w_1/w_2 \notin \mathbb Q##. Of course, since ##\mathbb Q \subset \mathbb R##, ##w_1/w_2 \notin \mathbb R## implies ##w_1/w_2 \notin \mathbb Q##.No, assuming that by ##nm## you mean to multiply ##n## and ##m#, it is an integer, not a 2-tuple of integers.

Okay do you take ##(n,m)##, the operation is then addition ?

binbagsss said:
Okay do you take ##(n,m)##, the operation is then addition ?
Yes.

Orodruin said:
Yes.
thank you for your help

## 1. What is an isomorphism?

An isomorphism is a mathematical concept that describes a relationship between two structures that preserves their essential properties. In simpler terms, it means that two objects are essentially the same, but may look different or have different names.

## 2. What is an example of an isomorphism in mathematics?

One example of an isomorphism is the isomorphism between the real numbers and the points on a number line. Both have the same structure and properties, but may be represented differently.

## 3. How do isomorphisms relate to periods in elliptic functions?

In the context of elliptic functions, an isomorphism describes the relationship between two elliptic curves that have the same periods. This means that they have the same frequency and amplitude, but may have different shapes or equations.

## 4. What is the significance of isomorphisms in mathematics?

Isomorphisms are important in mathematics because they allow us to understand and connect different structures and concepts. They also help us to identify patterns and similarities between seemingly unrelated objects.

## 5. Can isomorphisms be used to solve problems in real-world applications?

Yes, isomorphisms can be applied to real-world problems in various fields such as physics, computer science, and economics. For example, isomorphisms can be used to model and analyze complex systems, such as networks and biological processes.

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