Find Aut(Z) homework automorphisms

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In summary, to find Aut(Z), one must first determine the possible choices for automorphisms, which are defined by a(1) = 1 or a(1) = -1. This is because an automorphism must send a generator to a generator in order to be surjective. The mapping that sends x to its inverse is not always an automorphism, but it is in the case of Z. In general, the group must have certain conditions in order for this mapping to be an automorphism.
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Find Aut(Z).

I want to find Aut(Z) so I listed the generators of Z which are 1, and -1. Then from the hint in the book it says once I know where the generators are mapped then I can get where every other element gets mapped. So I said the only choices for the automorphisms are defined by a(1) = 1 or a(1) = -1.
How do I know that these are the only automorphisms? If I could say that an automorphism maps a generator to a generator that would help, but I don't really know that and can't prove it right now. Any thoughts?
 
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Automorphisms are homomorphisms, so if you know where an automorphism g sends a generator, you know where it sends everything (when the group has a single generator), since g(m) = g(m x 1) = m x g(1)*. Suppose g sends 1 to something other than 1 or -1. Well something must be sent to 1, since g is surjective. Suppose g(m) = 1. Then m x g(1) = 1, g(1) = 1/m, but 1/m is not even in Z unless m = 1 or m = -1. Check that the automorphisms you've defined are indeed automorphisms. In general, the mapping that sends x to its inverse is not an automorphism, but is in this case. See if you can figure out, in general, what conditions on the group must hold in order for such a mapping to be an automorphism.

* Actually, this looks a little confusing. Although when we speak of the group Z, we adopt additive notation (that is, we denote the group operation on x and y as x+y, whereas in a general group we would denote it as simply xy, i.e. in multiplicative notation). If we agree to use multiplicative notation when speaking of Z, then adding 1 to itself m times would be denoted as 1m, and adding x to y would be denoted xy. If we also agree to write elements of Z in bold, then what I wrote would look like this:

g(m) = g(1m) = g(1)m

In additive notation, it would look like:

g(m) = g(m x 1) or g(m1) = m x g(1) or mg(1)
 
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1. What is Aut(Z)?

Aut(Z) refers to the automorphism group of the integers, which consists of all bijective homomorphisms from the integers to itself. In other words, it is the set of all functions that preserve the algebraic structure of the integers.

2. How do I find automorphisms of Z?

To find automorphisms of Z, you can use the definition of an automorphism as a bijective homomorphism. This means the function must preserve addition, subtraction, and multiplication, and also be one-to-one and onto. You can also use specific properties of the integers, such as the fact that only +1 and -1 are their own inverses, to narrow down possible automorphisms.

3. What is the cardinality of Aut(Z)?

The cardinality, or size, of Aut(Z) is infinite. This is because there are infinitely many possible automorphisms of the integers, such as the identity function or functions that map every integer to its negative.

4. Can you give an example of an automorphism of Z?

One example of an automorphism of Z is the identity function, which maps every integer to itself. Another example is the function f(x) = -x, which maps every integer to its negative. Both of these functions preserve the algebraic structure and are bijective, making them automorphisms.

5. Why is finding automorphisms of Z important?

Finding automorphisms of Z can help us better understand the structure and properties of the integers. It can also be useful in other areas of mathematics, such as number theory and abstract algebra. Additionally, studying automorphisms can lead to insights and discoveries in other areas of science, such as cryptography or computer science.

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