Weird group isomorphism problem

In summary, the conversation discusses proving that the group Z/<(a,b)> is isomorphic to Z if gcd(a,b)=1 and finding generators for Z/<(a,b)>. The group <(a,b)> is a subgroup of Z, and for gcd(a,b)=1, every element in Z can be generated from a linear combination of a and b. However, this is not the case for Z/<(a,b)>, as it would result in a trivial group. The conversation also mentions that <(a,b)> may actually be written as <1>, but that does not provide a solution. The problem was brought up in a class and may be related to another problem involving ZxZ. Suggestions were made to
  • #1
y2kevin
6
0

Homework Statement


Show that the group Z/<(a,b)> is isomorphic to Z if gcd(a,b)=1. Find generators of Z/<(a,b)>.


2. Relevant information
Please note that the question is asking for Z/<(a,b)>, not ZxZ/<(a,b)>. I am having trouble understanding the meaning behind <(a,b)> as a subgroup of Z. My professor only gave the hint that <(a,b)> is cyclic and has an analogous case for ZxZ/<(a,b)> if gcd(a,b)=1.


The Attempt at a Solution


From my understanding, <(a,b)> should represent the set generated by a and b, ie, <(a,b)>={ n*a+m*b | n and m are integers }. However, for a and b s.t. gcd(a,b)=1, there are integers n and m such that n*a+m*b=1, in which case every element in Z can be generated from this linear combination.

This is clearly not the case since Z/<(a,b)> would be Z/Z={0}.

Does anyone have any idea what else <(a,b)> could be in this context

Thanks.
 
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  • #2
Normally, the group generated by a and b is written <a,b>. Also, gcd(a,b) is sometimes written (a,b). So maybe Z/<(a,b)> is actually Z/<1>. But that doesn't help either. Where did you find this problem?
 
  • #3
It came up during class following the problem ZxZ/<(7,37)>. Apparently these two problems share some similar characteristics.
 
  • #4
anyone?
 
  • #5
Have you tried working out the problem with ZxZ instead of Z? I'm pretty sure it should be ZxZ.
 

1. What is the "Weird group isomorphism problem"?

The "Weird group isomorphism problem" is a mathematical problem that deals with the isomorphism (or structural similarity) of two groups that are generated by certain specified elements. It was first introduced by mathematician Robert J. Wilson in 1976.

2. What makes this problem "weird"?

This problem is considered "weird" because it involves non-abelian groups, which have complex algebraic structures and can be difficult to study compared to abelian groups. Additionally, the specific elements used in the problem are chosen in a way that makes it challenging to determine whether the two groups are isomorphic or not.

3. What are some real-world applications of this problem?

The "Weird group isomorphism problem" has applications in cryptography, where it is used to design secure communication protocols. It also has connections to other areas of mathematics, such as topology and geometry, and has been studied in relation to certain physical systems in quantum mechanics.

4. Is there a solution to this problem?

As of now, there is no known algorithm or general solution to this problem. However, there have been some partial results and specific cases where the problem has been solved. It remains an active area of research in mathematics.

5. How does this problem relate to other problems in mathematics?

The "Weird group isomorphism problem" is related to other mathematical problems, such as the word problem and the isomorphism problem for groups. It also has connections to other fields of mathematics, such as group theory, combinatorics, and algebraic geometry. Its study has led to new ideas and techniques in these areas.

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