This past week I covered an abstract algebra course at Columbia, and I decided to prove that the alternating group is simple. I in fact did this in the same algebra class last year, but in the intervening months I almost entirely forgot how the argument goes. So I decided to write it up here while it’s still fresh in my mind. It’s a very nice – and fairly elementary – little application of some important ideas in group theory. In this post I’m going to give some background and explain the significance of the simplicity of , and in the sequel I will go through the proof.

### The Alternating Group

I am forced to assume that the reader is comfortable with basic group theory, but I’ll begin by reviewing some of the key ideas. Recall that the symmetric group is the group of permutations of symbols . A

*transposition*is a permutation which swaps exactly two symbols and leaves the others fixed; it is not hard to see that any permutation can be expressed as the product of transpositions. A permutation is said to be

*even*(respectively,

*odd*) if it can be written as the product of an even (respectively, odd) number of transpositions.

**Definition:** The *alternating group* is the subgroup of consisting of all even permutations.

is a normal subgroup of of index ; the objective of this series of posts is to prove that is simple for , meaning its only normal subgroups are itself and the trivial group. The significance of this property is that if a group has a normal subgroup then one can form the quotient group , and often one can infer properties of from properties of and . So simple groups are in a sense the “atoms” from which all other groups are built, though it should be noted that and alone do not uniquely determine .

### The Classification of Simple Groups

Classifying all finite simple groups was one of the great achievements of 20th century mathematics, and like many great mathematical achievements it went almost completely unnoticed by the rest of the world. The classification theorem asserts that all finite simple groups fit into a few infinite families (one of which is the family of alternating groups) with precisely 26 exceptions, the so-called *sporadic simple groups*. A shameless plug: when I was an undergraduate I did an REU project with Igor Kriz which involved making little computer games based on the sporadic simple groups; later we wrote a Scientific American article about them.

In any event, the classification program took decades to complete and spans thousands of pages written by dozens of mathematicians, and its completion seems to have essentially killed off finite group theory as an active area of research (though from what I understand there are lots of open problems in representation theory for finite groups). Given how monumental the effort was and how few people are still working in finite group theory, I worry that in a few decades all the experts will retire or die and there will be nobody left who understands the proof. It’s a good illustration of the principle that mathematicians tend to care much more about questions than answers.

### The Alternating Group and Galois Theory

Aside from their role in the classification program, the alternating groups play a crucial role in the theory of polynomial equations. Indeed, the very notion of a group was invented to understand the structure of solutions to polynomial equations, and the group is the star of the show.

Everyone learns in high school algebra that there is a formula for the roots of a quadratic equation :

Less well known is that there is also a cubic formula and quartic formula for degree three and four equations, respectively. These formulas date back to the 16th century, and it was a frustratingly difficult open problem to find a formula for the roots of a polynomial equation of degree five. It wasn’t until the 19th century that Abel and Galois independently realized that no such formula exists! Abel’s proof came first, but I don’t know what it was; Galois’ argument is the one that survived. Here is a brief sketch.

Galois’ key idea was to focus on the symmetries exhibited by the roots of a polynomial equation. More precisely, he considered their symmetries relative to the rational numbers; there are well-known techniques for finding rational roots of polynomials, so he was interested in the structure of the irrational roots. Let’s look at a couple examples:

- The roots of are , so you can get from one root to the other by multiplying by . Thus the cyclic group naturally exhibits the symmetries of the roots.
- The roots of are , , , and . Notice that you can cycle through the roots just by looking at powers of : , , , and . Thus the symmetries of the roots are given by the cyclic group .
- The roots of are and . The roots and are interchangeable, as are and , but over the rational numbers there is a sort of asymmetry between and . Thus the symmetry group is .

Of course, one can make all this precise using the language of field extensions. The upshot is that the symmetry groups help characterize what it means to find a formula for the roots of a polynomial equation. As in the example above, equations of the form have cyclic symmetry group . So if the quintic formula had the form , for instance, then the symmetry group could be decomposed into a part and a part corresponding to the fifth root and cube root, respectively. More precisely, a polynomial equation can be solved by radicals if and only if its symmetry group has a decomposition

where is a normal subgroup of and is cyclic. Groups with this property are said to be *solvable* due to the connection with solving equations.

Now, there exist polynomials of degree whose symmetry group is the full symmetric group (in fact there are many). contains as a normal subgroup with quotient , but once we have proved that is simple we will know that it is not solvable: it has no nontrivial normal subgroups whatsoever, let alone one with a cyclic quotient. This argument shows that there cannot be a general formula in the spirit of the quadratic, cubic, or quartic formulas, but it also shows even more: it gives you a criterion (solvability of the symmetry group) to determine when there is a formula for the roots of a specific polynomial.

J W Tanner

Jun 14, 2018@ 04:11:24Question:

When defining solvable group, you said G[i-1]/G[i] is cyclic, but Wikipedia says it’s Abelian !?

pwsiegel

Jun 15, 2018@ 20:15:24In the context of this post the group $G$ is finite since it is the Galois group of a polynomial, and for finite groups the definition of solvable using “cyclic” is equivalent to the definition using “Abelian” because every finite abelian group is the direct product of finite cyclic groups. It is true that the right generalization of “solvable” to infinite groups requires abelian rather than cyclic quotients, but I didn’t want to get into this nuance because these groups aren’t particularly relevant to the theory of polynomial equations.

J W Tanner

Jun 15, 2018@ 21:29:40Got it.

Thanks so much for clarifying.