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In many courses, say Linear Algebra, Group Theory, Topology, instructors often say that knowing that certaing objects are isomorphic, sometimes makes life easier because we can work with more friendly objects.

Is there any example of using isomorphism concept to solve a non trivial concrete problem?

I must say, I'm not looking on important or famous isomorphic objects, but on non trivial application of the concept of isomorphism used in order to solve another kind of problem.

HeMan
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    I suppose one could come up with something from ${\begin{pmatrix}a&b\-b&a\end{pmatrix}\mid a,b\in\Bbb R}\cong \Bbb C$. -- Or that the Möbius transforms under compositoin are isomorphic to $PGL(2,\Bbb C)$ – Hagen von Eitzen Jun 22 '17 at 05:26
  • @HeMan I think a big problem is what is a trivial problem and what is not trivial problem. – Michael Rozenberg Jun 22 '17 at 05:26
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    But my question is, basically, how do we use it? For example, that matrix example is a famous isomorphism, but I want to know how do we use that isomorphism to solve another problem. Because is a big problem of mathematics to classify objects, but I wonder, if, lets say, we already have classified all objects, then what happens next? Why do we want to classify objects? What does it give to us? – HeMan Jun 22 '17 at 05:32
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    There is an interesting example, I don't know if you count it as trivial, but here goes : I never managed to show the following result on its own "If the group $Inn(G)$ of inner automorphisms of $G$ is cyclic, then $G$ is abelian", I never saw how it worked, until I realized that I knew how to solve the (much easier in my opinion) problem "If $G/Z(G)$ is cyclic then $G$ is abelian". Once you know how to solve the second one, the first one follows easily via the isomorphism $Inn(G) \simeq G/Z(G)$. This is an "easy" example as it could fit in a basic algebra class, but (cont) – Maxime Ramzi Jun 22 '17 at 05:35
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    (Cont) it gives an idea of how isomorphisms can help : they can help you get closer to a situation you know and thus solve the original situation, thanks to the knowledge of another one – Maxime Ramzi Jun 22 '17 at 05:36
  • Not a mathematical example per se, but a real-life one from a previous career: minimizing the number of vias when routing a complex circuit board turned out to be isomorphic to the problem of register allocation by compilers. – amd Jun 22 '17 at 07:31
  • Pretty much the whole of C*-algebra theory is based on the Gel'fand representation theorem which establishes an isometric isomorphism and lets you switch between algebraic and analytic methods for their study. – postmortes Jul 03 '17 at 05:05
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    We classify objects so that we don't have to prove results from first principles for them each time. If we can show that they are isomorphic, we can apply all known results for that structure and we now know as much about that object as we know about that which it's isomorphic to. We don't have to study all 5-dimensional complex vector spaces because as vector spaces, they're all exactly the same. – rnrstopstraffic Jul 03 '17 at 05:05

2 Answers2

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Let's say you have a group of big matrices, maybe they are $10000 \times 10000$ matrices with lots of nonzero entries (so you are dealing with a subgroup of $\mathrm{GL}(10000,\mathbb{R})$). And you have to do a bunch of computations with this group. This is a lot of work to multiply these big matrices over and over.

BUT let's suppose you know this group of matrices is isomorphic to some group that is easier to deal with, like for example a cyclic group of order 50. It is really easy to do computations in this cyclic group (you can just do arithmetic modulo $50$). So if you know the isomorphism, you can do your computations in this cyclic group, and translate your results back to your original setting. Then you don't have to multiply the big ugly matrices together.

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A friend of mine once got out of a jaywalking ticket because he showed that the pattern of traffic lights at two adjacent intersections were isomorphic to a finite abelian group. From that point he used well-known results to prove that there were multiple elements in the group which allowed for safe crossing as opposed to only the one that was indicated by the "walk" light.

I doubt the judge understood the argument, but it worked. I'd call that practical and concrete!

rnrstopstraffic
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