Are U(20) and U(24) Isomorphic?

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In summary, to prove or disprove if U(20) and U(24) are isomorphic, we can compare the orders of their elements. In U(20), there are 8 elements with orders 1 or 2, while in U(24), there are 7 elements with order 1 or 2. Since they do not have the same number of elements with the same order, we can conclude that U(20) and U(24) are not isomorphic.
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Homework Statement



Prove or disprove U(20) and U(24) are isomorphic

Homework Equations





The Attempt at a Solution



U(20)={1,3,,7,9,11,13,17,19}
U(24)={1,5,7,11,13,17,19}

In U(20), Order(1)=1 Order(3)=4 Order(7)=4 Order(9)=2, Order(11)=2 Order(13)=4, Order(17)=4, Order(19)=2

In U(24), Order(1)= 1 , Order(5)=2 , ORder(7)=2, Order(11)=2 , Order(13)=2 , Order(17)=2, Order(19)= 2

Since each of the elements in each of the groups do not generate the same order for each elements in the opposite group, can't I conclude thatt U(24) and U(20) are not isomorphic?
 
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You can say that U(24) has 6 elements of order 2, while U(20) doesn't. So they certainly can't be isomorphic.

(By the way, you're missing 23 in U(24). So in fact U(24) has 7 elements of order 2, but this is immaterial.)
 
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1. What is an isomorphism in mathematics?

An isomorphism is a mathematical concept that describes a one-to-one correspondence between two mathematical structures, such as groups, rings, or vector spaces. It essentially means that the two structures are structurally identical, although they may look different.

2. What is the Isomorphisms problem?

The Isomorphisms problem is a computational problem that asks whether two given mathematical structures are isomorphic. In other words, it asks if there exists a function that maps one structure onto the other while preserving the structure of the elements. It is a fundamental problem in computer science and has applications in many fields, including cryptography and network security.

3. How is the Isomorphisms problem solved?

The Isomorphisms problem is a well-studied problem in computer science and there are various algorithms that can be used to solve it. The most common approach is to use a brute force method, where all possible mappings between the two structures are checked until an isomorphism is found. However, this approach can be computationally expensive, especially for large structures. Other approaches include graph isomorphism algorithms and group isomorphism algorithms.

4. What are the implications of solving the Isomorphisms problem?

Solving the Isomorphisms problem has significant implications in computer science and other fields. It can be used to verify the security of cryptographic systems, as an isomorphism between two structures would mean that the systems are equivalent. It can also be used to classify and analyze mathematical structures, which can lead to a better understanding of complex systems.

5. What are some real-life applications of the Isomorphisms problem?

The Isomorphisms problem has applications in various fields, including computer science, chemistry, and biology. In computer science, it is used in cryptography, network security, and data analysis. In chemistry and biology, it is used to identify and classify molecules and proteins, which can aid in drug discovery and understanding biological processes. It also has applications in social sciences, such as identifying patterns in social networks and analyzing cultural similarities between different groups.

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