Isomorphism and Generators in Z sub P

In summary, in order to prove that Aut(Z sub P) ≈ Z sub p-1, we must show that groups must preserve the operation, be 1-1, and be onto, also known as an isomorphism. Z sub p-1 has one less element and all the elements are the same except for one. Knowing what f does to x is enough to know where f sends any other element of the group. Additionally, Aut(Z_p) is isomorphic to the group of units of Z_p, and there is a natural isomorphism between the two. This can be proven by showing that the latter is cyclic.
  • #1
PennState666
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Homework Statement


Let P be a prime integer, prove that Aut(Z sub P) ≈ Z sub p-1


Homework Equations



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The Attempt at a Solution


groups must preserve the operation, be 1-1, and be onto and they can be called an isomorphism. Z sub p-1 has one less element in it so and all the elements in them are the same except for the one less element. Not sure what this tells me though. HELP ME FOR LINEARRRRR
 
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  • #2
Let f be an automorphism of Z_p, and x a generator for Z_p, so that <x>=Z_p. Explain why f is determined by what it does to x, i.e. why knowing f(x) suffices to to know where f sends any other element of the group.

Now think about whether (x is a generator) => (f(x) is a generator) is true. You will see that Aut(Z_p) is isomorphic to the group of units of Z_p -- there is a natural isomorphism. Think how you can prove that the latter is cyclic.
 

What is isomorphism with zprime?

Isomorphism with zprime is a mathematical concept that describes a relation between two sets of numbers. It is a one-to-one correspondence between the elements of two sets, such that the mathematical structure of the sets remains the same.

What is the significance of isomorphism with zprime?

Isomorphism with zprime is significant because it allows us to study and understand the properties and relationships of different sets of numbers by mapping them onto familiar sets, such as integers or real numbers. This helps us to simplify complex mathematical concepts and make connections between seemingly unrelated areas of mathematics.

How is isomorphism with zprime related to prime numbers?

Isomorphism with zprime is closely related to prime numbers because prime numbers are used to define zprime, which is a set of all positive integers that are relatively prime to a given number. Prime numbers also play a crucial role in determining the structure and properties of isomorphism with zprime.

What are some examples of isomorphism with zprime?

One example of isomorphism with zprime is the mapping between the set of integers and the set of even integers. Both sets have the same mathematical structure, and every integer can be mapped to a unique even integer and vice versa. Another example is the mapping between the set of rational numbers and the set of real numbers, which are both infinite sets with a one-to-one correspondence between their elements.

How is isomorphism with zprime used in other fields of science?

Isomorphism with zprime has applications in a wide range of scientific fields, including physics, chemistry, and computer science. In physics, it is used to study the symmetries of physical systems, while in chemistry it is used to understand the properties of molecules and their bonds. In computer science, isomorphism with zprime is used to optimize algorithms and data structures for efficient processing and storage of information.

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