Putting on your sock and then your shoe does not give the same result as putting on your shoe and then your sock. Adding 3 and then 5 gives the same as adding 5 and then 3. When the order of two operations does not change the result, we say they commute. The word mathematicians use for structures in which everything commutes is a tribute to Abel: abelian.

The last memoir

After proving that the general equation of the fifth degree has no formula, Abel wanted to know which equations do have one. In 1829, in one of the last papers he published in Crelle’s Journal, he studied a special family: equations in which all the roots can be obtained from one of them, $x$, by rational functions $\theta_1(x), \theta_2(x),\dots$ that commute with each other:

$$\theta_1(\theta_2(x))=\theta_2(\theta_1(x)).$$

Abel proved that these equations can always be solved by radicals. They came to be called abelian equations, and later the groups in which order does not matter, abelian groups.

The roots of unity

The most important example is the equation $x^n=1$. One of its roots is

$$\zeta=\cos\frac{2\pi}n+i\sin\frac{2\pi}n,$$

and all the others are powers of it: $\zeta^2,\zeta^3,\dots,\zeta^n=1$. The functions that take one root to another are the powers, $\theta_a(x)=x^a$, and they commute:

$$\theta_a(\theta_b(x))=(x^b)^a=x^{ab}=(x^a)^b=\theta_b(\theta_a(x)).$$

By Abel’s theorem, $x^n=1$ can be solved by radicals for every $n$. It is the same structure Gauss had used to construct the 17-gon: the vertices of the polygon are the roots of $x^{17}=1$.

Groups and commutativity

In modern language, a group is a set of operations that can be composed, undone and combined, always staying inside the set. The group is abelian when $a\circ b=b\circ a$ for any two elements.

Abelian examples are everywhere:

  • The integers under addition: $3+5=5+3$.
  • The rotations of a circle: turning $30°$ and then $50°$ is the same as turning $50°$ and then $30°$.
  • Clock arithmetic: on a 12-hour clock, 9 o’clock plus 5 hours gives 2 o’clock. We write $9+5\equiv2\pmod{12}$, and here too the order of the terms does not matter.

And there are groups that are not abelian. The symmetries of a triangle are the smallest example: rotating the triangle and then reflecting it does not give the same as reflecting and then rotating. The Rubik’s cube is another: turning the front face and then the top face does not have the same effect as the reverse. It was by studying non-abelian groups that Galois explained why the fifth degree has no formula.

Why commuting helps

The intuition behind Abel’s proof is that, in an abelian group, the operations can be performed one at a time, in any order, without one spoiling the work of another. Each step corresponds to extracting a root, and the roots can be extracted in sequence. In a non-abelian group that is too complicated, like that of the 120 permutations of five roots, the steps get in each other’s way, and there is no way to reach the end with radicals alone.

Exercise: in 12-hour clock arithmetic, solve 7 + x ≡ 3. Then solve 5 · x ≡ 1.

For the first, $x\equiv3-7=-4\equiv8$: from 7 o’clock, moving on 8 hours, you reach 3 o’clock. For the second, we test: $5\cdot5=25=2\cdot12+1$, so $x\equiv5$. The number 5 is its own inverse on the clock.

Exercise: on a square, do turning by 90° and reflecting in a diagonal commute?

Number the corners of the square 1, 2, 3, 4 anticlockwise, starting from the top right corner, and use the diagonal through corners 1 and 3. Turning $90°$ anticlockwise and then reflecting, corner 1 goes to position 2 and then to position 4. Reflecting first and then turning, corner 1 stays at position 1 and then goes to position 2. The results are different: the symmetries of the square do not form an abelian group. But the four rotations alone, by $0°$, $90°$, $180°$ and $270°$, commute with each other and form an abelian group.

Sources: Niels Henrik Abel, “Mémoire sur une classe particulière d’équations résolubles algébriquement”, Journal für die reine und angewandte Mathematik 4 (1829) · Israel Kleiner, A History of Abstract Algebra (Birkhäuser, 2007) · Abel biography, MacTutor.

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