Any example of functions are automorphism?

Question

I am looking for functions fulfilling $f(x+y) = f(x) + f(y)$ and $f(x*y) = f(x)*f(y)$.

I can only find $f(x)=x$, any more?

Hi Big John, I'm sorry but I don't understand your question. Since we are talking about functions, could you please state domain and codomain? Since we are talking about automorphisms which are isomorphisms of a mathematical object, you'd need to tell us which object we are talking about and which structure the isomorphisms are supposed to preserve. (And to whoever downvoted the question: Please explain in a comment why you downvote.) — Tim van Beek, Aug 08 '11 at 06:34
@Big John: I have tried to change a wording little bit (since many people point out that the question is not stated clearly enough), while I was trying to be as faithful to your original question as possible. Still, I cannot add the missing details, which were pointed out in comments (domain, codomain, requirements like bijectivity or continuity?) You should edit the question and add the missing details. — Martin Sleziak, Aug 08 '11 at 07:32
@Big John: Taking on a comment from a previous question of yours, it seems that you are trying to bite more than you can chew about basics of logic and set theory. These things don't always come naturally and easily, and often one has to destroy all intuition in order to develop a "correct" sense of mathematical intuition. Try finding a book or lecture notes online which cover these basics, and don't skip the details that you believe you know already. — Asaf Karagila, Aug 08 '11 at 08:11
my question is to find some example function of automorphism — Walker, Aug 08 '11 at 09:51

score 5 · Accepted Answer · edited Apr 13 '17 at 12:58

The question is not completely stated at this moment, but I thought that it would be better to move my comments to an answer. (Basically they're answering the question in the way I understood it, at least partially. Also, the comments section becomes unreadable after adding too many of them.)

You're asking about functions that simultaneously verify the equations $f(x+y)=f(x)+f(y)$ and $f(xy)=f(x)f(y)$. (You did not specify the domain and codomain, $\mathbb Q\to \mathbb Q$, $\mathbb R\to \mathbb R$ or $\mathbb C\to \mathbb C$ seems to be most plausible.

You should also include the conditions which you impose on $f$. (I guess at least bijectivity, since you ask about automorphisms in the second part of your question.)

You are correct that $f(x)=x$ is a solution. Obviously, $f(x)=0$ is a trivial solution for any of these cases, too. (It is not bijective.) If you want to work with functions from $\mathbb C$ to $\mathbb C$, then $f(x)=\overline x$ is a solution.

Here you can find the proof the the only automorphism of the field $\mathbb R$ is trivial, if this was what you intended to ask.

It might be interesting for you to know, that the first equation is called Cauchy functional equation. It is known that the only continuous solutions $f:\mathbb R\to\mathbb R$ are the functions of the form $f(x)=cx$. The proof can be found in some of the questions I linked bellow. So if you want such functions to be continuous, the second equation yields $c=1$ or $c=0$. Thus the only continuous solutions $\mathbb R\to \mathbb R$ are $f(x)=x$ and $f(x)=0$.

However, as the above proof shows, if you assume bijectivity and the second equation, the continuity is not needed.

Some related questions (and answers) which you could have a look at:
Field automorphisms of $\mathbb{Q}$ - shouldn't there be only one?
Are there any non-linear solutions of Cauchy’s equation ($f(x+y)=f(x)+f(y)$) without assuming the Axiom of Choice? (mathoverflow)
If $f(xy)=f(x)f(y)$ then show that $f(x) = x^t$ for some t

Any example of functions are automorphism?

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