Groups of Automporphisms - Aut(C) .... Anderson and Feil Ch. 24 ....

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In summary, Anderson and Feil state that Aut (C) is a group with only two elements, i.e. the identity automorphism and the complex conjugation map. However, if the axiom of choice is accepted, then this assertion is incorrect, for $\operatorname{Aut}(\Bbb C)$ has uncountably many elements.
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I am reading Anderson and Feil - A First Course in Abstract Algebra.

I am currently focused on Ch. 24: Abstract Groups ... ...

I need some help in understanding some claims in Chapter 24 by Anderson and Feil ... ...Anderson and Feil claim that \(\displaystyle \text{Aut} ( \mathbb{C} ) \)is a group with only two elements \(\displaystyle \{i, f \}\) ... ... where \(\displaystyle i\) is the identity automorphism and \(\displaystyle f\) is the complex conjugation map defined by \(\displaystyle f(a + bi) = a - bi\) ... ...

... can someone please help me to prove the assertion that \(\displaystyle \text{Aut} ( \mathbb{C} ) \)is a group with only two elements \(\displaystyle \{i, f \}\) ...

Peter
 
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Hi Peter,

Could you check back Anderson and Feil to see if they meant the automorphisms of $\Bbb C$ fixing $\Bbb R$?
 
  • #3
Euge said:
Hi Peter,

Could you check back Anderson and Feil to see if they meant the automorphisms of $\Bbb C$ fixing $\Bbb R$?
I am beginning to understand that whether the automorphisms of $\Bbb C$ fix $\Bbb R$ may be an issue ... but from my reading of Anderson and Feil they seem to imply that the the automorphisms of $\Bbb C$ actually fix \(\displaystyle \mathbb{R}\) anyway ... but you seem to be implying that this is not necessarily the case ...

The relevant text from Anderson and Feil is as follows:First the example that started me thinking:
https://www.physicsforums.com/attachments/6855
Now the above example involves a Galois Group ... and in this group \(\displaystyle \mathbb{R}\) is certainly fixed under Anderson and Feil's definition of a Galois Group ... which reads as follows:

https://www.physicsforums.com/attachments/6856
https://www.physicsforums.com/attachments/6857

Now, Example 47.3 refers to Exercise 24.14 which asserts that \(\displaystyle \text{Aut} ( \mathbb{C} )\) has only two elements ... note the exercise does not mandate the fixing of \(\displaystyle \mathbb{R}\) ... Exercise 24.14 reads as follows:View attachment 6858Hope that clarifies what Anderson and Feil have to say about the issue ... ...

Peter
 
  • #4
If Anderson and Feil reject the axiom of choice, then it would make sense for them to say $\operatorname{Aut}(\Bbb C)$ has two elements. However, if they do accept the axiom of choice, their statement about $\operatorname{Aut}(\Bbb C)$ would be incorrect, for in that case, $\operatorname{Aut}(\Bbb C)$ would be uncountably infinite.
 
  • #5
Euge said:
If Anderson and Feil reject the axiom of choice, then it would make sense for them to say $\operatorname{Aut}(\Bbb C)$ has two elements. However, if they do accept the axiom of choice, their statement about $\operatorname{Aut}(\Bbb C)$ would be incorrect, for in that case, $\operatorname{Aut}(\Bbb C)$ would be uncountably infinite.
Thanks Euge ...

Well ... I am surprised by that ... can you please explain how the Axiom of Choice makes such a difference in this matter ...

... it is not obvious to me how the Axiom of Choice makes such a difference ...

Peter
 
  • #6
With the axiom of choice, one can use the concept of a transcendence base to construct field automorphisms of $\Bbb C$ different from the identity and complex conjugation. More generally, it can be shown that the cardinal number of the field automorphisms of an algebraically closed field $\Bbb k$ is $2^{\text{card}(\Bbb k)}$. Since $\Bbb C$ has cardinal number $2^{\aleph_0}$, then with axiom of choice, the cardinal number of $\operatorname{Aut}(\Bbb C)$ is $2^{2^{\aleph_0}}$.

A natural question arises: how much choice is required for $\text{card}(\operatorname{Aut}(\Bbb C)) > 2$? I'm pretty certain you don't need the full AC for it to hold, but I don't know what the weakest choice assumption needs to be. Perhaps a logic expert like Evgeny.Makarov knows.
 
  • #7
Euge said:
With the axiom of choice, one can use the concept of a transcendence base to construct field automorphisms of $\Bbb C$ different from the identity and complex conjugation. More generally, it can be shown that the cardinal number of the field automorphisms of an algebraically closed field $\Bbb k$ is $2^{\text{card}(\Bbb k)}$. Since $\Bbb C$ has cardinal number $2^{\aleph_0}$, then with axiom of choice, the cardinal number of $\operatorname{Aut}(\Bbb C)$ is $2^{2^{\aleph_0}}$.

A natural question arises: how much choice is required for $\text{card}(\operatorname{Aut}(\Bbb C)) > 2$? I'm pretty certain you don't need the full AC for it to hold, but I don't know what the weakest choice assumption needs to be. Perhaps a logic expert like Evgeny.Makarov knows.
Thanks Euge ...

Still reflecting on this post ... need to read and think around this matter ...

Peter
 
  • #8
I would advise you not to think too deeply about this, for these issues are difficult to deal with, even for professional mathematicians. Just ignore the exercise as it is a faulty one.
 
  • #9
Euge said:
I would advise you not to think too deeply about this, for these issues are difficult to deal with, even for professional mathematicians. Just ignore the exercise as it is a faulty one.

Thanks for the advice/warning Euge ...

I will now leave this matter and move on to more Galois theory ...

Thanks for your help on this issue/problem ...

Peter
 

FAQ: Groups of Automporphisms - Aut(C) .... Anderson and Feil Ch. 24 ....

What is the definition of Automorphisms in group theory?

Automorphisms in group theory refer to isomorphisms from a group to itself, which preserve the group operation. In other words, an automorphism is a bijective function that maps the group elements to themselves while preserving the group structure.

What is the significance of studying Aut(C) in group theory?

Aut(C) is the group of all automorphisms of the complex numbers, and it has many important applications in different areas of mathematics, such as algebraic geometry and number theory. Understanding Aut(C) can provide insights into the structure and properties of complex numbers.

Can you give an example of an automorphism in Aut(C)?

An example of an automorphism in Aut(C) is the identity function, which maps each complex number to itself. Another example is the complex conjugation function, which maps a complex number z=a+bi to its conjugate z*=a-bi.

What is the order of Aut(C)?

The order of Aut(C) is infinite, as there are infinitely many ways to map the complex numbers to themselves while preserving the group structure. Each bijective function in Aut(C) is considered a different automorphism, making the group infinite.

How does the concept of Automorphisms relate to group actions?

In group theory, automorphisms can be seen as group actions on the group itself. This means that an automorphism is a way to transform the elements of a group while still maintaining the group structure. This perspective can provide a deeper understanding of the properties of automorphisms and their role in group theory.

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