8. The power of homomorphisms

When you remove the “homo” prefix you get morphism, which might suggest that a “morphism” is no longer between objects of the same (“homo”) type. This is not correct; a morphism is still actually defined between objects of the same “type” and is in fact a more abstract and arguably more complicated concept (coming historically much later).

8.1 Embeddings and quotient maps

Compare Homomorphism, Group homomorphism, and Embedding. The author doesn’t add anything original to those articles, but does fit a few concepts into book-specific language and gives unique visual examples. A few paragraphs at the bottom of pg. 160 regarding generating a full homomorphism from just a part of it doesn’t have a clear correspondence on Wikipedia, though it seems likely it has been invented before.

Intuitively, homomorphisms make “multiplication” (or in general the “operator”) work the same both before and after the application of the homomorphism function to all the involved elements. So if \(a = bc\) before, then \(ϕ(a) = ϕ(b)ϕ(c)\) after; it’s simply more compact to say this as \(ϕ(bc) = ϕ(b)ϕ(c)\). This meaning makes the concept of homomorphisms much more general than the concept of group homomorphisms. To make it clear we are talking about some general “operator” some articles (e.g. Group isomorphism) use special symbols like ⊙ (“Circled dot operator” in Unicode) or \(⋆\) (“Star operator”) rather than simply concatenating terms.

If the operator works the same on both sides of a map, that you can potentially perform the same operations on the second side of the map more efficiently. For example, there’s a map from the real numbers to the floating point numbers that’s approximately a homomorphism. Within the floating point numbers there are several types and levels of precision (single-precision, double-precision).

Through the rest of the book, the term “homomorphism” will be used only to mean Group homomorphism despite its more general definition.

8.1.1 Embeddings

Compare Embedding and Group isomorphism. In the second paragraph and a bit in the next section, the author categorizes all homomorphisms as being either (1) an embedding or (2) a quotient map. The article Group homomorphism would form an isomorphic partition of all homomorphisms by classifying them as injective or not (the language that we’ll prefer). That is, the author defines an “embedding” as a monomorphism (in the non-categorical sense).

The author claims that an embedding (i.e. monomorphism) is an isomorphism if its image fills the whole codomain. This is stated more generally in Group homomorphism § Image and kernel:

The kernel and image of a homomorphism can be interpreted as measuring how close it is to being an isomorphism.

In particular, the smaller the kernel is the closer the homomorphism is to being injective (it’s injective when the kernel is a single element). The large the image is, the closer it is being surjective (it’s surjective when the image fills the codomain).

8.1.2 Quotient maps

Compare Quotient maps and Kernel (algebra). The author’s argument in this section (“Observation” 8.2 through 8.4) sometimes seems to have holes. The whole thing can equivalently be boiled down to the following from Group homomorphism § Image and kernel:

The kernel of h is a normal subgroup of G. Assume \(u \in \operatorname{im}(h)\) and show \(g^{-1} \circ u \circ g \in \operatorname{im}(h)\) for arbitrary \(u, g\): $\(\begin{align} h\left(g^{-1} \circ u \circ g\right) &= h(g)^{-1} \cdot h(u) \cdot h(g) \\ &= h(g)^{-1} \cdot e_H \cdot h(g) \\ &= h(g)^{-1} \cdot h(g) = e_H, \end{align}\)$

Humans categorize; see Classification (a redirect of Categorization). To categorize something is closely related to the concept of Quotient maps. In both cases we take what are effectively different things (e.g. different traffic signs, or different elements of a set) and map them to the same thing (the equivalence class). This human habit is also the source of the wonderful Up to jargon. The benefit of categorization is that we can efficiently separate items we “care” about (that are “valuable”) from those we don’t care about. We often have to find some distinct aspect of worlds we care and don’t care about to act on them differently.

What makes a quotient map different from other kinds of equivalence relations is that it’s done on the elements of an algebraic structure (in this case, a group). Per Equivalence relation, these are called a congruence relation:

A congruence relation is an equivalence relation whose domain \(X\) is also the underlying set for an algebraic structure, and which respects the additional structure. In general, congruence relations play the role of kernels of homomorphisms, and the quotient of a structure by a congruence relation can be formed. In many important cases, congruence relations have an alternative representation as substructures of the structure on which they are defined (e.g., the congruence relations on groups correspond to the normal subgroups).

A congruence relation is a continuation of the theme that the “operator” works the same, as discussed above.

8.2 The Fundamental Homomorphism Theorem

The first isomorphism theorem states that the image of a homomorphism is isomorphic to the quotient by the kernel, or in short that \(im(f) \cong \frac{G}{ker(f)}\). It’s actually point 3. in Isomorphism theorems § Theorem A (groups). What’s interesting about it is that it applies to absolutely any homomorphism \(f\), and not just for group homomorphisms but to many other homomorphisms between algebraic structures of the same type.

Per Isomorphism theorems § Note on numbers and names, this theorem is also confusingly called the fundamental theorem on homomorphisms by some authors, and authors who call it the “fundamental theorem of homomorphisms” call a different theorem the first isomorphism theorem. Which is correct, or desirable?

The article fundamental theorem on homomorphisms currently describes a more general theorem than the one stated above, only requiring \(N \in ker(f)\). Notice in particular the distinction it creates with the statement:

Therefore, by setting \(N = ker(f)\), we immediately get the first isomorphism theorem.

We’ll follow this article in calling the simpler theorem the first isomorphism theorem (straying from VGT a bit). It turns out the author is not completely consistent either; see First Isomorphism Theorem - Group Theory Terminology.

In the short term we can essentially consider them synonyms, however, since we essentially always take \(N = ker(f)\).

8.3 Modular arithmetic

This section summarizes Fundamental theorem on homomorphisms § Integers modulo \(n\) in a visual way. It’s also discussed in the first two examples of Quotient group § Examples.

8.4 Direct products and relatively prime numbers

From Cyclic group § Additional properties:

If n and m are coprime, then the direct product of two cyclic groups Z/nZ and Z/mZ is isomorphic to the cyclic group Z/nmZ, and the converse also holds: this is one form of the Chinese remainder theorem. For example, Z/12Z is isomorphic to the direct product Z/3Z × Z/4Z under the isomorphism (k mod 12) → (k mod 3, k mod 4); but it is not isomorphic to Z/6Z × Z/2Z, in which every element has order at most 6.

The term “coprime” never appears in this book; the author uses the term “relatively prime” as an exact synonym (a synonym mentioned in Coprime). We’ll prefer “coprime” to “relatively prime” for brevity.

8.5 The Fundamental Theorem of Abelian Groups

What the author calls the “Fundamental Theorem of Abelian Groups” is known elsewhere as the Fundamental theorem of finite abelian groups (emphasis on finite), which is distinct from the Fundamental theorem of finitely generated abelian groups ((emphasis on finitely generated). From that article:

The fundamental theorem of finite abelian groups states that every finite abelian group \(G\) can be expressed as the direct sum of cyclic subgroups of prime-power order; …

This definition uses the term “direct sum” rather than “direct product” but for abelian groups these are equivalent, according to the article Direct product of groups:

In the context of abelian groups, the direct product is sometimes referred to as the direct sum, and is denoted \(G ⊕ H\). Direct sums play an important role in the classification of abelian groups: according to the fundamental theorem of finite abelian groups, every finite abelian group can be expressed as the direct sum of cyclic groups.

The terms “direct sum” and “direct product” are also equivalent in the context of abelian groups according to the article section Direct sum § Direct sum of abelian groups.

Notice that this theorem does not imply that every finite abelian group is isomorphic to a direct product of prime-order cyclic groups. For example \(C_4 \neq C_2 × C_2\) (see Exercise 7.18a). The theorem works in this case because \(C_4 = C_{2^2}\) (\(4\) is prime-power).

Direct sum vs. direct product

How do the terms “direct sum” and “direct product” differ in a context other than abelian groups? There doesn’t seem to be clear consensus on this topic. According to Direct sum § Examples:

In the case of groups, if the group operation is written as + the phrase “direct sum” is used, while if the group operation is written ∗ the phrase “direct product” is used. When the index set is infinite, the direct sum is not the same as the direct product since the direct sum has the extra requirement that all but finitely many coordinates must be zero.

The introduction to the article and the section Direct sum § Direct sum of abelian groups go into more detail on this topic. This definition seems to conflict with the article Direct sum of groups, which defines a direct sum in the context of two normal subgroups.

8.6 Semidirect products revisited

There are two versions of the symbol ⋊ (left and right). A helpful comment in Semidirect product § Notation:

One way of thinking about the \(N ⋊ H\) symbol is as a combination of the symbol for normal subgroup (\(◁\)) and the symbol for the product (\(×\)).

See also Automorphism and Automorphism group.