Proving Isomorphism of Complex Number Conjugate: General Algebraic Systems

[complexPHILOSOPHY]

I am working through this algebra book and some of the problems. The chapter this comes out of is General Algebraic Systems and the section is Isomorphisms. I am new to proofs and maths higher than calculus I so I am not sure if I am following the text or not. There aren't any solutions and this book is really old, out of print and not on the internet (temporary until I can purchase one of the texts already suggested to me).

Here is the problem:

Prove that the mapping $$a+bi\rightarrow a-bi$$ of each complex number onto its conjugate is an automorphism of the additive group of C.

$$(a+bi)\alpha=a-bi$$
$$(c+di)\alpha=c-di$$

$$[(a+bi)+(c+di)]\alpha=$$
$$[(a+c)+(b+d)i]\alpha=$$
$$(a+c)-(b+d)i=$$
$$(a-bi)+(c-di)=$$

$$(a+bi)\alpha+(c+di)\alpha$$

Since $$a-bi=c-di$$
Then $$(a-c)+(d-b)i=0$$

$$a=c$$ and $$b=d$$

$$a+bi=c+di$$

This is one-to-one right?

I think this is correct but I am not sure if it is even complete or if I am on the right track. If someone could help me, that would be great.

 

[Kreizhn]

Everything looks fine.

You've shown that the mapping preserves the binary operation under each respective group.

You've shown injectivity with

$$a=c$$ and $$b=d$$

Technically, you only need to show surjectivity, though it's a trivial exercise.

Other than that, everything looks perfectly fine.

 

[complexPHILOSOPHY]

Everything looks fine.

You've shown that the mapping preserves the binary operation under each respective group.

You've shown injectivity with

$$a=c$$ and $$b=d$$

Technically, you only need to show surjectivity, though it's a trivial exercise.

Other than that, everything looks perfectly fine.

An automorphism is an isomorphism from a set of elements onto itself, right? So, by demonstrating surjection as well, would I be illustrating a bijection? Do I need to show surjection for proof or were you suggesting it to help improve my understanding?

I appreciate your response. Also, since my book seems to be the only copy that was printed on the earth, I don't know how difficult the problems are considered. This problem seemed almost too simple (considering I have no idea what I am doing) so I am wondering if my book is very easy? Does this problem look pretty average or would you suggest more difficult problems to get a better intuitive feel for groups? I am a complete algebra newbie. This is my third day reading through the text, so bear with me.

Also, one further question. How long does the average person spend on each chapter when working through maths? I am completely self-taught all the way up to Calculus 1 and I have no experience learning math from anyone except books, so I am not sure what the pace of average students is compared to me.

I am curious because I will be taking Calculus II this semester and I am wondering if I will be slower than them.

 

[Kreizhn]

Well, from what I recall, the definition for a group isomorphism is

Definition The map $$\phi : G \rightarrow H$$ is an isomorphism if

1) $$\phi$$ is a homomorphism

2) $$\phi$$ is a bijection

where the group homomorphism preserves the binary operation mapped from G to H, and a bijective function is both injective and surjective. With that in mind, it's necessary, although relatively trivial to show that the complex conjugate is a one-to-one correspondence (i.e. Injective and Surjective).

It isn't usually until somewhat higher level courses that one is introduced to this kind of algebra (except possibly in an advanced classical algebra class), and so it's outstanding that you'd be learning this on your own.

As for this question in particular, it does indeed seem fairly rudimentary, but the difficulty of question you might want depends greatly on your background. Have you been introduced to the formal idea of proofs for abstract algebra? Is that the kind of thing you'd be interested in?

It's also somewhat hard to gauge what exactly can be meant by a chapter, and I suppose that also depends on the text that you're using. I too learned Calculus I and II via the textbook, and I found that doing a "chapter" daily was very sufficient. However, note that a university student doing Calc II typically only has this lecture 3 times a week (at least where I come from) and so a chapter daily would actually be an advanced pace.

I would imagine that this also depends on the amount of theoretical material presented in the course. When you studied Calc I were you introduced to the $$\epsilon\delta$$ notion of continuity? Cauchy Sequences? The notion of compact sets? Banach Spaces? Lipschitz, Topological, or metric definitions of continuity? I don't believe these are taught outside of advanced Calculus classes, but if this is the kind of thing you are learning, it wouldn't hurt to take a bit more time to ensure that you fully understand the material.