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The other day I was trying to figure out how to explain why isomorphisms are important. I pulled Boyer's A History of Mathematics off the bookshelf and was surprised to find that isomorphism isn't even listed in its index. The Wikipedia article on isomorphisms only gives two concrete examples.

There are many surprising, significant, classic isomorphisms. I'll refrain from giving examples. What are your favorites?

As usual, please limit yourself to one isomorphism per answer.

(Related: your favorite surprising connections in mathematics. But this question is looking for more concrete examples, particularly those that illustrate the power of the idea.)

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    $\begingroup$ This is a duplicate of math.stackexchange.com/questions/3173/… . $\endgroup$
    – Robin Chapman
    Commented Sep 2, 2010 at 18:57
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    $\begingroup$ I am surprised you're surprised that a book on the history of math doesn't mention isomorphisms. The target audience for a generic history of math treatise wouldn't have the technical background to appreciate the idea. The only reason such books can get away with mentioning Lebesgue integrals (I've seen it in some history of calculus books) is that it can be linked to something the readers have seen, namely the Riemann integral. Isomorphism, by comparison, is off the charts as far as typical students in a history of math class are concerned... (more) $\endgroup$
    – KConrad
    Commented Sep 2, 2010 at 19:02
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    $\begingroup$ It is unclear how a list of people's favorite isomorphisms is going to help you explain isomorphisms. $\endgroup$
    – KConrad
    Commented Sep 2, 2010 at 19:03
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    $\begingroup$ @stankewicz I'm not sure why this question would draw that comment while "What is your favorite counterexample?" does not. I think both "counterexample" and "isomorphism" are words that everyone's familiar with and which are pretty easy to explain in terms of lower-level primitives (like propositions and quantification on the one hand, or bijections and relations on the other). $\endgroup$
    – Jason Orendorff
    Commented Sep 2, 2010 at 19:48
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    $\begingroup$ In view of the insightful answers given and since I disagree with a practice where overlapping groups of high point users vote to close the same question both at MU and MO, I am voting to reopen. $\endgroup$
    – Victor Protsak
    Commented Sep 3, 2010 at 3:16

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Here is an example that Mel Hochster used to explain the notion of isomorphism to a group of talented high school students. I was one of the course assistants rather than one of the students, but I'm sure the insight was at least as valuable for me as for them.

Consider the following game. I'll write down the numbers 1 through 9 on a sheet of paper, and you and I will take turns selecting numbers from the list (crossing off each number once it has been selected). The winner is the first person to have chosen exactly three numbers which add up to 15. For example if I selected 9, 6, 2 and you selected 3, 8, 1, 4 then you would win because 3 + 8 + 4 = 15.

The interesting thing is that this game is isomorphic to tic-tac-toe. I don't know what I precisely mean by that, but I can explain why it is true. Simply consider a 3 x 3 magic square:

4 9 2

3 5 7

8 1 6

The rows, columns, and diagonals all add up to 15, and moreover every way of writing 15 as the sum of three numbers from 1 to 9 is represented. When you choose a number, draw an X over it; when I choose a number, circle it. Tic-tac-toe!

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    $\begingroup$ "...this game is isomorphic to tic-tac-toe. I don't know what I precisely mean by that, but I can explain why it is true." The trees of the two games are isomorphic. Your explanation gives the isomorphism in terms of the labels on the edges. $\endgroup$
    – Jason Orendorff
    Commented Sep 3, 2010 at 12:36
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English and French are isomorphic.

Stronger. They are both trivial.

See this paper by Mestre, Schoof, Washington, and Zagier for a short proof.

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    $\begingroup$ Cantonese and Mandarin are isomorphic, with the isomorphism given by written Chinese. $\endgroup$
    – John Stillwell
    Commented Sep 2, 2010 at 23:48
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    $\begingroup$ But the isomorphism is not unique: you have both the "traditional" and "simplified" isomorphisms :) $\endgroup$
    – José Figueroa-O'Farrill
    Commented Sep 3, 2010 at 0:43
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    $\begingroup$ It's much cooler that Japanese is isomorphic to the free group on 46 generators. $\endgroup$
    – Sheikraisinrollbank
    Commented Sep 3, 2010 at 9:15
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The set of positive reals under multiplication is isomorphic to the set of reals under addition, which is the isomorphism underlying the operation of a slide rule. This is the only isomorphism I can think of important enough that its explicit (approximate) values used to be published in 1000-page books. The positive reals under multiplication is also a standard pedagogical example of an interesting one-dimensional abstract real vector space, where there is some content to verifying the axioms. (The other standard example is the reals with addition given by $x+y-1$ and multiplication by scalar $a$ given by $ax + 1 - a$.)

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    $\begingroup$ I find the parenthetical example contorted, but here is a related one that's important in the theory of formal groups: the multiplicative group of a field, $(x,y)\mapsto xy,$ where $x,y\ne 0,$ is isomorphic to the group with the group law $(x,y)\mapsto x+y+xy,$ where $x,y\ne -1.$ $\endgroup$
    – Victor Protsak
    Commented Sep 3, 2010 at 1:28
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The most striking example of an isomorphism I remember seeing as an undergraduate was when John Conway visited and gave his famous talk on rational tangles. Being able to unknot a seemingly hopelessly tangled pair of skipping ropes by manipulating rational numbers was an amazingly concrete demonstration of what it meant for structures to be isomorphic.

For those who don't know what I'm talking about, I think there's a video online somewhere.

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The Cantor pairing function is the function $p(a,b)= (a+b)(a+b+1)/2 + b$, a polynomial bijection between the pairs of natural numbers and individual numbers. Thus, it is a bijection or isomorphism of the sets $\mathbb{N}\times\mathbb{N}$ and $\mathbb{N}$. Using such a function, one may easily deduce that the set of rational numbers is countable, and more generally, that a countable union of countable sets is countable.

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I nominate the Chinese Remainder Theorem, in the form of an isomorphism of a ring of residues with a cartesian product ring. This isn't "profound" mathematics, but simply unpacking it (with construction of the underlying idempotents) should convince students that algebraic structure has "content". I recall a conversation about the analogue for polynomials in one variable over a finite field, in which my side was really stating that if you understand the original CRT in the correct way, this is no sweat at all.

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I'm a fan of the isomorphism between $PSL_2 (F_7)$ and $GL_3 (F_2)$ (two nice descriptions of the simple group of order 168). Years ago, Richard Guy asked me if I knew an explicit map, and I didn't. But recently one was given in the Math Monthly:

MR2572107 Brown, Ezra; Loehr, Nicholas Why is ${\rm PSL}(2,7)\cong{\rm GL}(3,2)$? Amer. Math. Monthly 116 (2009), no. 8, 727--732.

The paper is also available from Brown's website:

http://www.math.vt.edu/people/brown/doc.html

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The elliptic modular function
j(τ) = q-1 + 744 +196884q + ... (q=e2πiτ)

This is an isomorphism from elliptic curves (such as C/(1,τ)) to the complex plane.

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The set of 7-tuples of binary trees is isomorphic to the set of binary trees. For the correct definition of "isomorphic" this is a surprising non-trivial result.

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One of my current favourites can be found on Peter Cameron's blog. Take a countable model $(M,E)$ of enough (axioms of) set theory. Symmetrize the relation $E$ to obtain a graph. This graph is the random graph (Rado's homogeneous universal countable graph).

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The whole subject of non-commutative geometry arises from extending to non-commutative algebras the isomorphism that exists between commutative $\mathrm{C}^*$-algebras and locally compact Hausdorff topological spaces.

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  • $\begingroup$ This might be my favorite example with actual mathematical content. $\endgroup$
    – Paul Siegel
    Commented Sep 2, 2010 at 22:03
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    $\begingroup$ Hmm. I thought (in my limited understanding) that the real point is that the isomorphism is a natural isomorphism at the level of functors... And I think that the answer given by Tracy makes a nice slogan but is perhaps eliding over some historical details (IMHO) $\endgroup$
    – Yemon Choi
    Commented Sep 2, 2010 at 22:55
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    $\begingroup$ I'm no expert either, of the subject or of its history, and I'm sure the slogan is an overstatement depending on how it is interpreted. I just meant that this deep equivalence at the level of axioms, or isomorphism of categories, is the reason that it even makes sense to speak of "non-commutative" topological spaces, since the algebraic axiomatization of the category of locally compact Hausdorff topological spaces includes an axiom of commutativity that can be relaxed. $\endgroup$
    – Tracy Hall
    Commented Sep 3, 2010 at 0:04
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    $\begingroup$ The correct statement is indeed that the Gelfand transform, regarded as a functor from the category of commutative C*-algebras to the category of locally compact Hausdorff spaces, is an equivalence of categories. And strictly speaking Connes' noncommutative geometry program is not about "extending the isomorphism" - so far as I know there is no sensible way to extend the Gelfand transform to a functor from the category of all C*-algebras to some other category. Rather, it is about using C*-algebras as proxies for spaces and extending geometric tools to the noncommutative case. $\endgroup$
    – Paul Siegel
    Commented Sep 3, 2010 at 6:17
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    $\begingroup$ That being said, the Gelfand-Naimark theorem really is the key idea that gets Connes' program started and inspires many of the relevant constructions. $\endgroup$
    – Paul Siegel
    Commented Sep 3, 2010 at 6:19
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The De Rham Isomorphism.

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  • $\begingroup$ It's a beautiful one. Mysterious too (the period conjecture). $\endgroup$
    – AFK
    Commented Sep 2, 2010 at 22:50
  • $\begingroup$ Nowadays it's better if answers contain more content than this, e.g., an explanation, link, description, etc. $\endgroup$
    – David White
    Commented Feb 13 at 15:49
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The isomorphism between $SL_2(\mathbb{C})$ and the universal covering of the special Lorentz group $SO^+(1, 3)$ is definitely nifty in my opinion. ("Coincidences" between Lie groups are another good source of examples.)

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I like the isomorphism between a finite abelian group and its "Cartier" dual (not the bidual!) precisely because it's non-canonical. But I don't think it makes a good example for explaining isomorphism to non-mathematicians.

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A calculator made using wire and logic gates with electrons flowing through it should be considered isomorphic to a calculator made using tubes and physical gates with water flowing through it as long as the underlying structure (the schematics for each calculator) is the same.

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    $\begingroup$ I agree that this is an important example of isomorphism, although probably not of much interest to mathematicians. I hear engineers discussing several classes of non-electronic phenomena in the language of electronic circuit diagrams, which often provides a convenient isomorphic setting for performing calculations. $\endgroup$
    – Tracy Hall
    Commented Sep 2, 2010 at 22:08
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I should say I'm fond of the Thom isomorphism, but I still find the contents rather mysterious.

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  • $\begingroup$ Nowadays it's better if answers contain more content than this, e.g., an explanation, link, description, etc. $\endgroup$
    – David White
    Commented Feb 13 at 15:51
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How about the Fourier transform as an isomorphism between the Hilbert space $L^2$ of quadratically integrable complex-valued functions on the unit interval and the Hilbert space $\ell^2$ of sequences of complex numbers the sum of the squares of whose norms is finite?

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  • $\begingroup$ I'd argue that this is not so much interesting in the category of Hilbert spaces and bounded linear maps, as in categories of Hilbert spaces with structure and bounded linear maps preserving that structure $\endgroup$
    – Yemon Choi
    Commented Sep 2, 2010 at 23:54
  • $\begingroup$ Which additional "structure" do you have in mind that gets preserved, besides the Hilbert-space structure? $\endgroup$
    – Michael Hardy
    Commented Sep 3, 2010 at 0:46
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    $\begingroup$ Group actions (this is a version of, or manifestation of, the fact that Z and T are dual abelian groups). I was also alluding to the fact that any separable Hilbert space is isomorphic to $\ell^2$, and yet in many instances this doesn't simplify the theory or the problems -- the reason we study lots of different Hilbert spaces is because they are attached to other structure (e.g. reproducing kernel Hilbert spaces) $\endgroup$
    – Yemon Choi
    Commented Sep 3, 2010 at 5:31

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