Automorphism groups and determing a mapping

In summary: So you can just take your formula for phi(x), phi(x)=ax mod 50, and take a=1 which is phi(1) and then solve for phi(x).
  • #1
irebat
7
0
1. Suppose that Ø:Z(50)→Z(50) is an automorphism with Ø(11)=13. Determine a formula for Ø(x).
this is the problem I am getting, its chapter 6 problem 20 in Gallian's Abstract Algebra latest edition (you can find it on googlebooks) Am i wrong in thinking there's something wrong with the problem?
I think this because arent both 11 and 13 generators of U(50) because of coprimeness, am i missing something big? or is the book just giving me a bum problem?

edit: i have a final exam tommorow and the teacher stressed this chapters homework, please help!
 
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  • #2
hey guys I asked my TA and he walked me through it.

I seemed to understand when he explained it to me but when i got home for the life of me i can't seem to remember what he did.

this is the work i wrote down, can someone explain to me the thought process here?

---------------

If this is respect to addition, note that
11n = 1 (mod 50)
==> 11n = 1 - 100 = -99 (mod 50)
==> n = -9 = 41 (mod 50).

Therefore, 41 * 11 = 1 (mod 50)
==> Ø(1)
= Ø(11 + 11 + ... + 11) [41 times]
= 41 * Ø(11), since Ø is a homomorphism
= 41 * 13 (mod 50)
= -9 * 13 (mod 50)
= -117 (mod 50)
= 33.

Therefore for any x in Z_50, we have
Ø(1) = 33 (mod 50) ==> Ø(x) = 33x (mod 50).
 
  • #3
The thought process is that phi(x) is probably of the form a*x mod 50. You want to solve for a. You have that phi(11)=13, so you want a*11=13 mod 50. You would then solve for a just like in the real numbers. If you could find an inverse to 11 in Z_50 you could just multiply both sides by that inverse. So a*11*(11)^(-1)=a=13*(11)^(-1)=1. To actually find 11^(-1) mod 50 your TA used a clever trick to conclude 11^(-1) mod 50 is 41. I.e. 11*41=1 mod 50.
 

What is an automorphism group?

An automorphism group is a mathematical concept that describes a group of symmetries or transformations that can be applied to an object while preserving its structure and properties. It is a fundamental concept in group theory and is often used in the study of algebraic structures.

What is the role of automorphism groups in determining a mapping?

Automorphism groups play a crucial role in determining a mapping between mathematical objects. They provide a way to understand the structure and properties of objects and how they are related to each other. By studying the automorphism group of an object, one can determine the possible mappings that preserve its structure.

How are automorphism groups different from symmetry groups?

Although both automorphism groups and symmetry groups deal with symmetries and transformations, they are different concepts. Symmetry groups focus on the symmetries that preserve the geometric shape of an object, while automorphism groups deal with the symmetries that preserve the algebraic structure and properties of an object.

Can automorphism groups be used in real-world applications?

Yes, automorphism groups have many real-world applications. They are used in computer science, cryptography, and physics, among other fields. For example, in computer science, automorphism groups are used to study the symmetries of data structures and algorithms, which can lead to more efficient and robust software.

How can I determine the automorphism group of a given object?

The process of determining the automorphism group of an object can vary depending on the object and its properties. In general, one can start by looking at the structure and properties of the object and then identifying the transformations that preserve them. From there, one can build the automorphism group by combining these transformations in different ways.

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