# More Abstract-Homomorphism&quotient

• TheForumLord
In summary, the conversation discusses various problems related to quotient groups and homomorphisms. It covers proving the existence and uniqueness of a cyclic subgroup of order n in the additive quotient group Q/Z, determining the containment of finite subgroups of Q/Z, finding all homomorphisms from Z/nZ to Q/Z, and finding all homomorphisms from Q/Z to Z. The conversation also includes helpful hints and explanations on how to approach each problem.
TheForumLord

## Homework Statement

The last parts of the problems were:
1. prove that for each natural n, the additive quotient group Q/Z contains a one and only subgroup of order n and that sub-group is cyclic.
2. let G,H be two finite sub-groups of Q/Z. prove that G is contained in H if and only if
o(G)|o(H).
3. find all homomorphisms from Z/nZ to Q/Z.
4. find all homomorphisms from Q/Z to Z...

TNX for all the helpers...

none.

## The Attempt at a Solution

none...

For number 4, consider a nonzero element of Q/Z. You can represent it by a rational n/m with n and m integers. If you add n/m to itself m times, you get 0 in Q/Z. What does that tell you about the image of n/m under the homomorphism?

For the first one, try finding a more familiar group that's isomorphic to Q/Z (hint: think complex)

Dick said:
For number 4, consider a nonzero element of Q/Z. You can represent it by a rational n/m with n and m integers. If you add n/m to itself m times, you get 0 in Q/Z. What does that tell you about the image of n/m under the homomorphism?

Let n/m+Z be a nonzero element of the quo. group. the order of this element if m (if we assume that gcd (n,m)=1) . the image of n/m under the homo. has to be an element in Z of order m. But in Z, which element has order m?? how come this is the only homo. possible?

Help is needed :(

TNX !

If phi is your homomorphism, it doesn't really say phi(n/m) has order m. It says phi(n/m)*m=0. The only integer satisfying that is 0. So phi(n/m)=0, for all elements of Q/Z. That's why it's the only homomorphism possible.

so the homo is:
each and every element in Q/Z goes to 0?

TNX

TheForumLord said:
so the homo is:
each and every element in Q/Z goes to 0?

TNX

What else could it be?

right...Tnx a lot! Hope you'll be able to help me in my next thread too...

I'm pretty lame in all the Homomorphism theorems...

TNX again

Just start thinking about it. If you know what the image of the element '1' in Z/nZ is, then you know the rest of the homomorphism. What are the possibilities?

The image of "1" in Z/nZ (which is of course "n" or "0" ) can be each and every one of the generators of the sub-groups from order n...I've proved that there's only one sub-group of order n in Q/Z and it's cyclic...Each one of it's generators are from the form i/n... Am I on the right track?

TNX a lot!

Z/nZ={0,1,2,...n-1}. '1' doesn't mean 0 or n. It means 1. But yes, every element of Z/nZ has order n, so the image of an element i under the homomorphism has to satisfy phi(i)*n=0 in Q/Z. That makes it one of the i/n, sure.

TNX a lot man...You were very helpful!

## 1. What is an abstract homomorphism quotient?

An abstract homomorphism quotient, also known as a group quotient or quotient group, is a mathematical concept that describes the "quotient" or "division" of one group by another. It involves finding a new group that contains elements from the original group, but with certain elements identified as equal or equivalent.

## 2. How is an abstract homomorphism quotient different from a regular quotient?

An abstract homomorphism quotient is different from a regular quotient because it involves not only dividing groups, but also preserving the structure of the original group. This means that the new group formed by the quotient will also have the same operations and properties as the original group, making it a more specialized and powerful tool in mathematics.

## 3. What is the significance of abstract homomorphism quotients in mathematics?

Abstract homomorphism quotients have many applications in mathematics, particularly in algebra and group theory. They allow mathematicians to study the structure and properties of groups in a more efficient and organized way, and they also provide a way to classify and compare different groups.

## 4. How do you construct an abstract homomorphism quotient?

To construct an abstract homomorphism quotient, you first need to identify a normal subgroup of the original group. This is a subgroup that is invariant under conjugation by elements of the original group. Then, you can use the elements of the normal subgroup to form cosets, which will become the elements of the new quotient group. Finally, you define the operations and properties of the quotient group based on the cosets and the original group's operations.

## 5. Are there any real-world applications of abstract homomorphism quotients?

While abstract homomorphism quotients may seem abstract and theoretical, they do have real-world applications. For example, they can be used in cryptography to create secure encryption algorithms, and they are also important in physics and chemistry for understanding the symmetries and conservation laws of physical systems.

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