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Comment by soVeryTired | Hacker News Reader

I really dislike the standard definition of a group - that it's a set S together with a binary operation * such that a bunch of properties hold. The definition doesn't build intuition, and doesn't motivate the introduction of the concept of "group".

For me, a group is the set of isomorphisms of a graph. If you expand the definition of "graph" a little to include continuous spaces, that is sufficient to define all groups. And yet, this nice, intuitive definition of a group might show up at the end of a course in group theory - if you're lucky.

It really is a shame how much intuition is stripped from mathematics teaching in the name of formalism.

[0] https://en.wikipedia.org/wiki/Frucht%27s_theorem

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If something is intuitive only at the end of 4-5 month long course, it is not intuitive and I would not show it during any sort of "introduction to...". Definition requiring former explanation of a whole graph theory would be something I would put at the end... of an article about graph theory. I cannot make many assumptions about my readers, and CS magna cum laude graduate is definitely not one of them (why would they need the article then?).

You don't need a massive amount of graph theory. "Rearrange the labels of the graph so that the new graph looks the same as the old one" isn't exactly arcane, is it?

My humble experience shows me that this might be too much of a digression for many people. Graphs as a visual example of e.g. dependencies or data structures are fine. But if you try to explain something with them you assumes that reader had some prior exposition to graph theory and has some intuitions already. Since I cannot assume they have these intuitions I would have to build them with other examples and the digression could be longer than a paragraphs.

Abstractions are something that works great when you worked enough with some class of specific problems, that your brain notices common parts on its own. If you try to rush it... you get another tutorial when author is enthusiastic and positive and readers feel ashamed and stupid that they "couldn't get it". I'd rather avoid that. If they get foundations, play with them and gain some confidence, they can move on to more challenging and generic definitions.

Aren't Cayley graphs covered in most courses on group theory?

Since I'm not an expert in graph theory, especially the arcane theory of "continuous graphs", your definition is unintuitive and excessively formal to me. How is that better? Abstractions are abstract. Almost no one learns purely by abstraction. Any definition must be accompanied by examples, and always is when taught.

Grouoa are taught in the context of addition/subtraction or permutations or symmetry groups, far more intuitive that graphs.

I prefer thinking about group theory as the study of the symmetric group of order n. Then to make sense of the case of n = infinity, you introduce the group axioms.

Eventually, you can justify the study of permutations using the fact that if you take any algebraic structure and consider the group of automorphisms, you get a group.

> For me, a group is the set of isomorphisms

Well, then you are missing the entire (additive) group of integers... But you are right, in that automorphisms are the most important example, and they also play a huge theoretical role (as representations of abstract groups). This is what gives the group theory its importance in math and physics.

The additive group of the integers is the set of isomorphisms of a line with evenly spaced points and a direction.

Alternatively, take a directed graph with:

V = The integers

E = {x, y | x - y = 1}

and that graph has the same isomorphism group.

Replace the nodes of that graph with asymmetric graphs, and the resultant undirected graph again has the same isomorphism group.

Again, this is only what is called 'a representation.' (The problem with trying to use one as a substitute for the abstract definition is that there usually are many different representations.)

I think it would be very difficult to prove things about groups using that definition. For instance, it's not obvious to me how one would even define a subgroup in this context.

I'd guess it would be a subset of the set of isomorphisms that fixes a certain subset of the graph. But maybe I'm wrong.

I don't disagree that working straight from the graph-theoretic definition might make things harder. My complaint is that maths is taught as formal definition -> theorems. What I would like to see is intuitive definition -> formal abstraction of intuition -> theorms.

In my maths degree I spent far too long asking myself why is the definition of this thing this way?

There's a saying in math: Analysts understand what they're talking about but find it hard to proving things - while algebraists don't know what they're talking about but find it easy to prove things.

I find this to be true, at least at the introductory level. Once you get to topology you forget what you're talking about, but that's the structural ("algebraic") view of math resurfacing. Maybe you're right though - perhaps there should be a progression in algebra from concrete -> abstract the same way there's a progression from concrete (real analysis) -> abstract (topology)

I wrote a thesis very closely related on Cayley-graphs, and I was not aware of this theorem.

That is a really helpful theorem for intuition. Holy shit!

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