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I recently asked a question regarding why homomorphisms and isomorphisms are important. The best answer to that question was actually a comment, which referred me to Brian M. Scott's answer here: https://math.stackexchange.com/a/242370/115703

That answer was mind blowingly insightful for me. I finally began to understand why someone would care about homomorphisms, and why the "kernel" might actually be called a kernel. Revelation upon revelation. Frisson all over.

Why isn't this sort of an explanation easy to find in algebra textbooks though? (I am reading Dummit and Foote, and Rotman) Shouldn't this be the first thing a textbook says. Example:

Say we are interested in studying the structure of the odd and even numbers. If we look at it from the perspective of the set $\mathbb{Z}$, then we are likely to carry on a lot of extra baggage since $\mathbb{Z}$ has more structure in it than just "odd and even". What if studied the structure $\mathbb{Z}/2\mathbb{Z}$ instead? Well, it would be useful then to have some sort of a mapping between $\mathbb{Z}$ and $\mathbb{Z}/\mathbb{2Z}$, since we are really studying integers, in a "reduced structure" setting. What sort of mappings might we be interested in... (ellipsis for a better way to explain the argument between where I have left off, and where I am going to, which eludes me right now) -- so the concept of a homomorphism.

However, we might also be interested in asking how a homomorphism between $\mathbb{Z}$ and $\mathbb{Z}/2\mathbb{Z}$ changes/preserves the structure of $\mathbb{Z}$. For instance, we might be curious about which elements in $\mathbb{Z}$ essentially become the "same" in $\mathbb{Z}/2\mathbb{Z}$...--so, the concept of a kernel. On the other hand, which elements retain some sense of "difference"...--so, the concept of image.

An isomorphism is just a homomorphism which preserves detail exactly -- i.e. it doesn't collapse any elements in $\mathbb{Z}$ into the "same" element in $\mathbb{Z}/2\mathbb{Z}$..-- which is why its kernel is just the identity.

Actually, maybe instead of viewing homomorphisms and isomorphisms as structure preserving maps between groups, we should view them as generators of groups? i.e. given some group, and we construct something that is a homomorphism in order to explore a new group related to the old group, given that homomorphism?

A lot of what I have written is very "soft" and not formally fleshed out. Some parts are outright skipped over (ellipsis) because I still lack the wisdom to explain it well. Regardless, the point is that, when first learning these topics, in order to understand the definitions well, it would be very illuminating to read the "big picture" behind all the details that are about to follow.

Is there an abstract algebra text that provides this sort of illumination? Ideally, it such a text would also contain all the necessary proofs for formally defining a topic, but perhaps that is asking for too much?

Even more ideally, such a text would deal with most of abstract algebra (at least groups, rings and fields), but that might be asking for too much, again. So, perhaps recommendations can be split up into categories depending on which aspect of abstract algebra they tackle in particular.

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    $\begingroup$ Usually what happens is that the textbook provides the skeleton, technical details, proofs, a few examples and lots of exercises. What you are describing is usually the job of the instructor to convey. A textbook could, in theory, provide all these intuitions in writing along with the material, but firstly it would blow up to unmanageable length, and secondly there is no guarantee it would give you what you want. Look at your situation: first you struggled with the material, made the effort yourself, then came across Brian's post and only then you were enlightened. Cont. $\endgroup$ – guest Oct 13 '15 at 20:19
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    $\begingroup$ Do not underestimate how much your own previous struggles factored in the enlightenment. Of course this does not mean it would be harmful for a textbook to provide this intuition for each and every new topic (length aside), but it is not clear how beneficial it will be if you haven't previously took pains to comprehend, conjure and reject ideas and intuitions, and so on. This is why it is best, in my opinion, to have a textbook providing the structure and problems to come up against, and the instructor to guide you with revealing insight, revealing precisely because of the previous struggle. $\endgroup$ – guest Oct 13 '15 at 20:22
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    $\begingroup$ Finally, do not forget that, brilliant as Brian's description may be, it is still an 'interpretation' or a 'viewpoint' of the material. By feeding a viewpoint, the texbook necessarily confines you to that way of thinking. There may be other ways to think, or intuit, about the material and these can lead to diverging paths. The experienced instructor can easily switch viewpoints, give ones more compatible with your vision and so on. The textbook's viewpoint will just stand there forever. $\endgroup$ – guest Oct 13 '15 at 20:26
  • $\begingroup$ @guest Maybe you want to write up your comments as an answer? :) Also: the method you outline in your comments feels great, but kind of impossible if you are an undergrad dealing with 4 other courses at once. How would you respond to that? $\endgroup$ – user89 Oct 19 '15 at 17:37
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    $\begingroup$ @guest Well, I feel like if I didn't pick this up from my instructor by attending class, then he must think that I should have come up with these insights on my own? This is likely the case, but given all my other responsibilities (e.g. other coursework), is it reasonable to expect that I should struggle through and come up with all insights myself? I would have loved to do that (doing math for the sake of math!), but that is not the reality of undergraduate education. $\endgroup$ – user89 Oct 20 '15 at 21:11
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I think I am now finally in some position to answer this question, but my answer will still be updated with time.

Let me begin by stating that recognizing isomorphisms and homomorphisms as definitions in their own right is non-trivial, and historically took much time to develop. From Stillwell's Elements of Algebra (note that Stillwell is also the author of the equally fabulous Mathematics and its History):

The concepts of isomorphism and homomorphism emerged only gradually in algebra, being observed first for groups around 1830, for fields around 1870 and for rings around 1920. In his memoir on the solvability of equations, Galois [1831] implicitly analysed groups by means of homomorphisms...

The first to use the term "isomorphism" was Jordan, in his Traite des Substitutions [1870], the first textbook on group theory...Jordan used the word "isomorphism" for both isomorphisms and homomorphisms, but distinguished between the two by calling them "isomorphismes holoedriques" and "isomorphismes meriedriques" respectively.

I don't know why Jordan chose the words he did for those concepts, but I recently put up a question to figure out why.

It is curious to note that "homomorphism" is first used in Stillwell's Elements of Algebra in the context of rings, where:

...the structure of a ring can often be elucidated by a homomorphism onto a simpler ring - recall how we learned about the structure of $\mathbb{Z}$ in Chapter 2 by mapping it onto $\mathbb{Z}/n\mathbb{Z}$...

I suppose Stillwell's book is a little odd in general because it takes the reader from rings to fields, and then finally groups.

Anyway, let us now shift our attention to Pinter's A Book of Abstract Algebra. In Pinter's book, Groups are introduced in Chapter 3, but homomorphisms are first discussed as a concept in their own right in Chapter 14. Here, Pinter tell us:

...This notion of homomorphism is one of the skeleton keys of algebra, and this chapter is devoted to explaining it and defining it precisely.

It's not difficult to define homomorphisms precisely, but the fact that Pinter considers this concept to be deep, and its value not immediately apparent from its definition alone gives us hope. He continues (empahsis retained from text):

The function $f: \mathbb{Z} \mapsto P$ [where $P = \{e, o\}$] which carries every even integer to $e$ and every odd integer to $o$ is...a homomorphism from $\mathbb{Z}$ to $P$.

...Now, what do $P$ and $\mathbb{Z}$ have in common? $P$ is a much smaller group than $\mathbb{Z}$, therefore it is not surprising that very few properties of the integers are to be found in $P$. Nevertheless, one aspect of the structure of $\mathbb{Z}$ is retained absolutely intact in $P$, namely the structure of the odd and even numbers. (The fact of being odd or even is called the parity of integers.) In other words, as we pass from $\mathbb{Z}$ to P we deliberately lose every aspect of the integers except their parity; their parity alone (with its arithmetic) is retained, and faithfully preserved.

Another example will make this point clearer. Remember that $D_4$ is the group of the symmetries of the square.

enter image description here

Now, every symmetry of the square either interchanges the two diagonals here labeled 1 and 2, or leaves them as they were. In other words, every symmetry of the square brings about one of the permutations of the diagonals.

...$S_2$ is a homomorphic image of $D_4$. Now, $S_2$ is a smaller group than $D_4$, and therefore very few of the features of $D_4$ are to be found in $S_2$. Nevertheless, one aspect of the structure of $D_4$ is retained absolutely intact in $S_2$, namely the diagonal motions. Thus, as we pass from $D_4$ to $S_2$ we deliberately lose every aspect of plane motions except the motions of the diagonals; these alone are retained and faithfully preserved.

A final example may be of some help...The most basic way of transmitting information is to code it into strings of $0$s and $1$s, such as $0010111$, $1010011$, etc. [called "binary words"]...The symbol $\mathbb{B}_n$ designates the group consisting of all binary words of length $n$, with an operation of addition [bitwise OR]...

Consider the function $f: \mathbb{B}_7 \mapsto \mathbb{B}_5$ which consists of dropping the last two digits of every seven-digit word. This kind of function arises in many practical situations: for example, it frequently happens that the first five digits of a word carry the message while the last two digits are an error check. Thus, $f$ separates the message from the error check.

It is easy to verify that $f$ is a homomorphism, hence is a homomorphic image of $\mathbb{B}_7$. As we pass from $\mathbb{B}_7$ to $\mathbb{B}_5$, the message component of words in $\mathbb{B}_7$ is exactly preserved while the error check is deliberately lost.

Need I say more? No, I do not need to say more, but I will: Pinter's book is awesome. I surveyed a few abstract algebra books over the past month, and this is the only one I found that gives importance to the concept of isomorphism/homomorphism (quite common in algebra texts), and helps give the reader a flavour of why the concept is important (very rare in algebra texts).

If you find more books that pass the "how well do they explain homomorphisms?" litmus test, please share!


Something extra, to complete the picture of homomorphisms: it is a very good strategy, when studying something complex, to identify and label only the most salient features of the structure we are interested in. Here is some justification for this from a problem solving perspective, in a lecture by Edsger Dijkstra (the Dijkstra in Dijkstra's algorithm; just check out the goat, wolf and cabbage example, which is the very first one): https://www.youtube.com/watch?v=0kXjl2e6qD0

More from Edsger Dijkstra: http://www.cs.utexas.edu/~EWD/

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  • $\begingroup$ Awesome answer! Did you learn more insights from Pinter's book after writing this answer? If so, I'd be delighted to hear them :) $\endgroup$ – Prism Sep 8 '16 at 23:34
  • $\begingroup$ @Prism It has been a long while, so some of the insights might have evaporated, but I'll try and think about it. $\endgroup$ – user89 Sep 10 '16 at 19:02
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This Chinese book 抽象代数基础(第二版) is also full of insights https://www.amazon.cn/dp/B014MBTMCO , the author 丘维声 is good at explaining things. Throughout the book he gives motivations for many of the definitions and theorems and gives plenty of insightful examples. No idea if there's any English translation available though. Perhaps someone should translate it.

When introducing the concept of homomorphisms he gives the example of projecting the image of a 3D object on three perpendicular planes (two walls and the floor). It's worth noticing that just by looking at the three different projected images one can figure out the original shape of the object. Projecting from 3D to 2D just means getting rid of one of the coordinates. (it's very similar to the function f:B7↦B5 which consists of dropping the last two digits of every seven-digit word, described above by user89.) For example the projection upon the Oxy plane can be noted mathematically this way:

proj((x,y,z))=(x,y)

It is a (homo)morphism because it is a linear map:

proj((x1,y1,z1)+(x2,y2,z2))=(x1,y1)+(x2,y2)=proj((x1,y1))+proj((x2,y2))

proj(a(x,y,z))=proj((ax,ay,ax))=(ax,ay)=a*(x,y)=a*proj((x,y,z))

Just as the projections of a teacup on three different planes revealed the size and shape of the teacup, so the different homomorphisms images of a group reveal the group's structure.

Thus homomorphisms are an important tool for studying the structures of groups and of many other algebraic/mathematical sets/structures, and as we've seen with this 3D projection example, homomorphisms are a very natural concept inspired from real-life situations.

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I'm turning my comment on the original post into an answer:

The texts "Visual Group Theory" by Nathan Carter, "An Illustrated Theory of Numbers" by Martin H. Weissman and "Indra's Pearls: The Vision of Felix Klein" by Mumford, Series, and Wright have really helped to develop my intuition. And they are visually stunning.

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