[abstract algebra] Isomorphic group of units

In summary, the conversation discusses the proof of an isomorphism between two groups, \mathbb Z_{nm}^\times and \mathbb Z_n^\times \oplus \mathbb Z_m^\times. The conversation suggests using Euler's totient function to prove that both groups have the same number of elements, but struggles with constructing the isomorphism. A potential solution is proposed by considering the function f: \mathbb Z_{nm}^\times \rightarrow \mathbb Z_n^\times \oplus \mathbb Z_m^\times, which maps units to units and can be proven to be surjective using the Chinese Remainder Theorem. However, it is important to note that homework helpers should not provide
  • #1
nonequilibrium
1,439
2

Homework Statement


Given that gcd(n,m)=1, prove that [itex]\mathbb Z_{nm}^\times = \mathbb Z_n^\times \oplus \mathbb Z_m^\times[/itex].

Homework Equations


/

The Attempt at a Solution


I can prove both groups have the same amount of elements (using Euler's totient function), but I can't figure out how to prove the isomorphism. One way would be to construct the isomorphism, but I can't seem to find one.
 
Physics news on Phys.org
  • #2
Perhaps consider function [itex] f: \mathbb Z_{nm}^\times \rightarrow \mathbb Z_n^\times \oplus \mathbb Z_m^\times [/itex] such that f(a) = (a mod n, a mod m). Since if [itex]a \in \mathbb Z_{nm}^\times[/itex] then a is relatively prime to mn, so there is an integer solution x,y to the equations [itex]ax +mny =1[/itex]. Taking the equation, you can re-write it to [itex]mny = 1 - ax[/itex], which means [itex]ax[/itex] is congruent 1 modulo m, and congruent 1 modulo n. So actually a has an inverse mod m and mod n (in this case x). So f maps units to units. You can prove surjectiveness by Chinese Remainder Theorem, and since sets are finite this would imply bijectivity.

/edit reworded slightly.
 
Last edited:
  • #3
Thank you!
 
  • #4
As a reminder to all homework helpers, we're supposed to help the student solve a problem, not to do it for them. Barre's post is a demonstration of exactly what not to do. Unfortunately, it's too late and the original poster has already received the complete solution to his homework problem. :frown:
 

1. What is an isomorphic group of units in abstract algebra?

An isomorphic group of units in abstract algebra is a group that has the same structure and properties as another group, but with different elements. This means that the two groups are essentially the same, but the elements are represented differently.

2. How do you determine if two groups are isomorphic?

Two groups are isomorphic if they have the same structure and properties. This can be determined by finding a bijective function, or a one-to-one and onto function, between the elements in the two groups. If such a function exists, then the groups are isomorphic.

3. Can an isomorphic group of units have a different number of elements?

No, an isomorphic group of units must have the same number of elements as the original group. This is because the groups have the same structure and properties, and therefore, the same number of elements is required to maintain this equivalence.

4. What is the significance of an isomorphic group of units in abstract algebra?

Isomorphic groups of units are important in abstract algebra because they allow for the comparison and analysis of different groups. By finding isomorphic groups, we can understand the properties and structure of one group by looking at the properties and structure of another group.

5. How are isomorphic groups of units used in mathematics and other fields?

Isomorphic groups of units are used in various areas of mathematics, including group theory, number theory, and algebraic geometry. They are also used in other fields, such as computer science, to study and analyze different structures and systems.

Similar threads

  • Calculus and Beyond Homework Help
Replies
5
Views
785
  • Linear and Abstract Algebra
Replies
13
Views
892
  • Calculus and Beyond Homework Help
Replies
2
Views
1K
  • Calculus and Beyond Homework Help
Replies
10
Views
1K
  • Calculus and Beyond Homework Help
Replies
5
Views
1K
  • Calculus and Beyond Homework Help
Replies
26
Views
2K
  • Calculus and Beyond Homework Help
Replies
3
Views
1K
  • Calculus and Beyond Homework Help
Replies
7
Views
1K
  • Calculus and Beyond Homework Help
Replies
9
Views
2K
  • Calculus and Beyond Homework Help
Replies
8
Views
1K
Back
Top