Count & Describe Automorphisms of Z_3 (Set of Integers Modulo 3)

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In summary, the conversation is about finding the number of Z_3 vector space automorphisms of Z_3(alpha) and describing them. The question is unclear on what alpha represents, but it is later defined as a root of x^2+1. Z_3(alpha) is a 2-dimensional vector space over Z_3 and the solution involves finding invertible 2x2 matrices over Z_3.
  • #1
erraticimpulse
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Please bear with me as I don't have latex. This is a homework question I have and I don't even know if it makes sense:

How many Z_3 (set of integers modulo 3) vector space automorphisms of Z_3(alpha) are there? Describe them.

I'm not sure if alpha is supposed to be the root of some polynomial or just an element outside of Z_3. I know that Z_3 is isomorphic to GF(3) (the Galois field of order 3). Any help would be much appreciated.
 
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  • #2
If you don't know what alpha is, then we have no chance of knowing what it is. The only person who can help you here is you: find out from your book, notes, the question sheet, what alpha is.
 
  • #3
Okay well, I figured it out. Alpha was defined a few pages earlier as a root of x^2+1 (a polynomial with coeff's in Z_3[x]). Thanks for pointing out what should have been obvious to me.
 
  • #4
So Z_3[alpha] is just a 2 dimensional vector space over Z_3, so we're just looking at the invertible 2x2 matrices over Z_3. What answer did you get?
 

1. What are automorphisms of Z_3?

Automorphisms of Z_3 are functions that preserve the structure of the set of integers modulo 3, also known as the cyclic group Z/3Z. This means that the automorphisms must satisfy the property that f(a+b) = f(a) + f(b) and f(ab) = f(a)f(b) for all integers a and b.

2. How many automorphisms are there for Z_3?

There are a total of 2 automorphisms for Z_3. These are the identity function, which maps every integer to itself, and the function that maps 0 to 0, 1 to 2, and 2 to 1. This function is often denoted as f(x) = x^2.

3. What is the significance of automorphisms in Z_3?

Automorphisms play an important role in group theory, as they are essentially symmetries of a group. In the case of Z_3, the automorphisms represent the different ways in which the elements of the group can be rearranged while preserving the group structure. This is important in understanding the properties and behaviors of Z_3 and other groups.

4. How can automorphisms of Z_3 be visualized?

Automorphisms of Z_3 can be visualized as rotations and reflections on a circle. Since Z_3 is a cyclic group, it can be represented as a circle with 3 equally spaced points labeled 0, 1, and 2. The automorphisms then correspond to rotating or reflecting the circle in different ways while keeping the points in the same positions.

5. Can automorphisms be applied to other sets of integers modulo n?

Yes, automorphisms can be defined for any set of integers modulo n, denoted as Z/nZ. The number of automorphisms for Z/nZ will depend on the value of n, but they will always include the identity function and a function similar to f(x) = x^2 as seen in Z_3. These automorphisms also have similar properties and significance as those in Z_3.

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