(Z/10557Z)* as Abelian Groups using Chinese Remainder Theorem

In summary, by using the Chinese Remainder Theorem and the fact that 10557 = 3^3 . 17 . 23, the group (Z/10557Z)* is isomorphic to (Z/27Z)* x (Z/17Z)* x (Z/23Z)*, which can be written as C18 x C16 x C22 where Cn represents the Cyclic group order n. To write this as Cn1 x Cn2 x Cn3 s.t. n1 divides n2 divides n3, we can use the invariant factor decomposition method by computing the elementary divisor decomposition of the group and finding the largest factor first, followed by the remaining factors until all the elementary div
  • #1
RVP91
50
0
If I was to try to work this out I would use the Chinese Remainder Theorem and since 10557 = 3^3 . 17 . 23
end up with (Z/10557Z)* isomorphic to (Z/27Z)* x (Z/17Z)* x (Z/23Z)* isomorphic to C18 x C16 x C22 where Cn represents the Cyclic group order n.

How would I then write this as Cn1 x Cn2 x Cn3 s.t. n1 divides n2 divides n3?
 
Physics news on Phys.org
  • #2
RVP91 said:
If I was to try to work this out I would use the Chinese Remainder Theorem and since 10557 = 3^3 . 17 . 23
end up with (Z/10557Z)* isomorphic to (Z/27Z)* x (Z/17Z)* x (Z/23Z)* isomorphic to C18 x C16 x C22 where Cn represents the Cyclic group order n.

How would I then write this as Cn1 x Cn2 x Cn3 s.t. n1 divides n2 divides n3?

Good start. You're looking for the invariant factor decomposition of the group--let's start out by computing its elementary divisor decomposition. Use the Chinese remainder theorem again on ##\mathbb{Z}/18 \mathbb{Z} \times \mathbb{Z}/ 16 \mathbb{Z} \times \mathbb{Z}/ 22 \mathbb{Z}## to write them as a product of cyclic groups with prime power order. This gives you the elementary divisors of the group.

From there, take a look at this link to convert to the invariant factor decomposition. Basically you find the largest factor first by choosing the largest prime power for each of the distinct primes and then multiplying them together. You then proceed similarly until you've used up all the elementary divisors. The link I gave has some examples, too.

Hope that helps!
 

1. What is the Chinese Remainder Theorem?

The Chinese Remainder Theorem is a mathematical theorem that states that if we have a set of congruences with relatively prime moduli, we can find a unique solution to the system of congruences. In other words, it helps us solve systems of linear congruences with different moduli.

2. How does the Chinese Remainder Theorem apply to Abelian groups?

In the context of Abelian groups, the Chinese Remainder Theorem allows us to break down a group into its prime power components and then use those components to construct a new group with the desired properties. This is useful in solving problems involving Abelian groups, as the Chinese Remainder Theorem provides a systematic way of constructing new groups from given ones.

3. What is the significance of "(Z/10557Z)*" in the context of Abelian groups?

The notation "(Z/10557Z)*" represents the multiplicative group of integers modulo 10557. In other words, it is the set of all integers between 0 and 10557 that are relatively prime to 10557, with the operation being multiplication modulo 10557. This group is useful in the context of Abelian groups as it allows us to work with a finite set of integers and still have the properties of a group.

4. How can we use the Chinese Remainder Theorem to classify Abelian groups?

The Chinese Remainder Theorem can be used to classify Abelian groups by their prime power components. By breaking down a group into its prime power components, we can determine the structure of the group and classify it accordingly. This is particularly useful in understanding the properties and behavior of Abelian groups.

5. Can the Chinese Remainder Theorem be extended to non-Abelian groups?

Yes, the Chinese Remainder Theorem can be extended to non-Abelian groups, but with some restrictions. In non-Abelian groups, the prime power components may not commute with each other, so the theorem may not hold in its original form. However, there are variations and generalizations of the theorem that can be applied to non-Abelian groups.

Similar threads

Can (Z/10557Z)* be written as Cn1 x Cn2 x Cn3 with n1 dividing n2 dividing n3?
    • Linear and Abstract Algebra
Replies
1
Views
2K
MHB Understanding the Chinese Remainder Theorem for $\mathbb{Z}^{\times} _{20}$
    • Linear and Abstract Algebra
Replies
5
Views
1K
Isomorphism types of abelian groups
    • Linear and Abstract Algebra
Replies
2
Views
4K
Find and Test Primes using the Chinese Remainder Theorem and Binary Search
    • Linear and Abstract Algebra
Replies
2
Views
5K

Hot Threads

Recent Insights

Back
Top