Problem: Abstract Algebra (3rd Edition) Dummit and Foote, Chapter 3, 3.1, pg 85

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Problem: Abstract Algebra (3rd Edition), by Dummit and Foote, Chapter 3, 3.1, pg 85:

The problem is stated as follows:

Define $f: R^* \to \{1,-1\}$ by letting $f(x)$ be $\frac x{|x|}$. Describe
the fibers of $f$ and prove that $f$ is a homomorphism.

Dummit and Foote define fibers only in the context of homomorphisms. As in we need to first prove that $f$ is a homomorphism, and then we proceed to deduce the fibers over the elements belonging to the image.

Definition of a "fiber" in Dummit and Foote: "If $F$ is a homomorphism from $G$ to a group $H$ recall that the fibers of $F$ are the sets of elements of $G$ projecting to single elements of H".

The answer is straightforward, but what is bothering me is the following:

1. Are fibers only associated with homomorphisms? What if a certain function from a group $H$ to $G$ is not a homomorphism, I may still be able to define a similar idea of the "fiber" right?

2. Also, why do I need that the function be a homomorphism in order to define the concept of a fiber?

3. Can I first deduce the fibers associated with the given function, and THEN prove that the given function is a homomorphism?

Best Answer

Dummit and Foote define fibers only in the context of homomorphisms.

This assertion is incorrect. Looking at my copy (Second Edition), pages 1 and 2:

We shall use the notation $f\colon A\to B$ or $A\stackrel{f}{\to}B$ to denote a function $f$ from $A$ to $B$

[...]

For each subset $C$ of $B$, the set $$f^{-1}(C)=\{a\in A\mid f(a)\in C\}$$ consisting of the elements of $A$ mapping into $C$ under $f$ is called the preimage or inverse image of $C$ under $f$. For each $b\in B$, the preimage of $\{b\}$ under $f$ is called the fiber of $f$ over $b$. [emphasis added].

This happens in the book before the introduction of the term "homomorphism", whose first occurrence seems to be in page 37.

By the way, I found the definition of fiber through the complicated and arcane process of checking the index of the book.

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