# 8. The power of homomorphisms#

When you remove the â€śhomoâ€ť prefix you get morphism, which might suggest that a â€śmorphismâ€ť is no longer between objects of the same (â€śhomoâ€ť) type. This is not correct; a morphism is still actually defined between objects of the same â€śtypeâ€ť 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 doesnâ€™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 doesnâ€™t have a clear correspondence on Wikipedia, though it seems likely it has been invented before.

Intuitively, homomorphisms make â€śmultiplicationâ€ť (or in general the â€śoperatorâ€ť) work the same both before and after the application of the homomorphism function to all the involved elements. So if \(a = bc\) before, then \(Ď•(a) = Ď•(b)Ď•(c)\) after; itâ€™s simply more compact to say this as \(Ď•(bc) = Ď•(b)Ď•(c)\). 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 â€śoperatorâ€ť some articles (e.g. Group isomorphism) use special symbols like âŠ™ (â€śCircled dot operatorâ€ť in Unicode) or \(â‹†\) (â€śStar operatorâ€ť) 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, thereâ€™s a map from the real numbers to the floating point numbers thatâ€™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 â€śhomomorphismâ€ť 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 weâ€™ll prefer). That is, the author defines an â€śembeddingâ€ť 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 Â§ 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 (itâ€™s injective when the kernel is a single element). The large the image is, the closer it is being surjective (itâ€™s surjective when the image fills the codomain).

### 8.1.2 Quotient maps#

Compare Quotient maps and Kernel (algebra). The authorâ€™s argument in this section (â€śObservationâ€ť 8.2 through 8.4) sometimes seems to have holes. The whole thing can equivalently be boiled down to the following from Group homomorphism Â§ 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 â€ścareâ€ť about (that are â€śvaluableâ€ť) from those we donâ€™t care about. We often have to find some distinct aspect of worlds we care and donâ€™t care about to act on them differently.

What makes a quotient map different from other kinds of equivalence relations is that itâ€™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 â€śoperatorâ€ť works the same, as discussed above.

## 8.2 The Fundamental Homomorphism Theorem#

The fundamental theorem on homomorphisms states that the image of a homomorphism is isomorphic to the quotient by the kernel. It actually applies to many structures other than groups. The author provides some great visual examples but nothing original relative to Wikipedia.

Per Isomorphism theorems Â§ Note on numbers and names, this theorem is also confusingly called the first isomorphism theorem by some authors, but authors who call it the â€śfundamental theorem of homomorphismsâ€ť call a different theorem the first isomorphism theorem. The authors who call it fundamental are generally earlier, and the title â€śfundamental theorem of homomorphismsâ€ť is currently not ambiguous, and therefore â€śfundamentalâ€ť should be preferred.

## 8.3 Modular arithmetic#

## 8.4 Direct products and relatively prime numbers#

From Cyclic group Â§ Additional properties:

If

nandmare coprime, then the direct product of two cyclic groupsZ/nZandZ/mZis isomorphic to the cyclic groupZ/nmZ, and the converse also holds: this is one form of the Chinese remainder theorem. For example,Z/12Zis isomorphic to the direct productZ/3ZĂ—Z/4Zunder the isomorphism (kmod 12) â†’ (kmod 3,kmod 4); but it is not isomorphic toZ/6ZĂ—Z/2Z, in which every element has order at mostÂ 6.

## 8.5 The Fundamental Theorem of Abelian Groups#

What the author calls the â€śFundamental Theorem of Abelian Groupsâ€ť is known elsewhere as the fundamental theorem of *finite* abelian groups, which is distinct from the fundamental theorem of *finitely generated* abelian groups.

## 8.6 Semidirect products revisited#

See also Automorphism.