Motivation behind studying Alternating Groups

Question

Going through "A Book of Abstract Algebra" by Charles Pinter now.

At the end of Chapter 8 Permutations of a finite set, he says that:

"The set of all even permutations in $S_n$ is a subgroup of $S_n$. It is denoted by $A_n$ and is called the alternating group on the set $\{1,2,\dots, n\}$"

What is interesting about even permutations that make them worth labelling and studying? It seems like a whole class of object being pulled out of thin air. Pinter doesn't do a good job of motivating the study of these objects.

Note: I have also watched this video on "Visual Group Theory" covering Alternating groups, and it doesn't cover motivation too much either (rather, just saying that such objects exist), although I find it interesting that alternating groups can represent platonic solids.

Answer 1

Even permutations appear often enough in practice to warrant particular attention (for instance, a permutation of the coordinate axes of $\Bbb R^n$ is orientation preserving iff it is even).

Here is another important and more theoretical reason: For $n\geq5$, they are simple groups, which is to say, they have no normal subgroups (aside from the trivial subgroup and itself).

Simple finite groups play a role for finite groups similar to what primes part for natural numbers; in a sense each finite group is "composed" of a collection of simple groups. More specifically, given a finite group $G$, and a maximal (proper) normal subgroup $N$, the quotient $G/N$ is simple. Then we can look at $N$ and take a maximal normal subgroup $M$, and the quotient $N/M$ is again simple.

Keep going like this until you get to the trivial group, and you have a so-called composition series of $G$. No matter which maximal normal subgroup you choose at each step, the simple groups that appear as quotients will be the same (up to isomorphism, and the order can change). These composition series are an important characteristic of a group, and possibly most famously appear in the proof of the insolubility of the general quintic (insolubility happens first in degree $5$ because $S_5$ is the first symmetric group with a non-abelian simple group, $A_5$, as a quotient in its composition series).

So simple groups are important in general, and the most available simple groups are the prime order cyclic groups and the alternating groups (for $n\neq 4$). So they are important.

Answer 2

I think the basic reason is that sub-structures of algebraic structures are almost always important and this is possibly the first non-trivial (and relatively unfamiliar) example we encounter in Group Theory (the dihedral groups have subgroups of rotations which have an immediate geometric interpretation). The significance of this will grow on you as you study more, but if you are uninterested in subobjects you will not get very far with abstract algebra.

The fact that $A_n$ is a subgroup is interesting in itself (ie that the parity of a permutation is well-defined and is preserved by the group operation). Properties which are preserved in this or similar manner become very important for mathematics. The sign of a permutation may seem a trivial thing, but for something apparently trivial it comes up surprisingly often. For example, quadratic reciprocity is related to this idea.

The Alternating Groups are also normal subgroups of the Symmetric Groups and for $n\ge 5$, $A_n$ is the only non-trivial normal subgroup of $S_n$.

The $A_n$ are also simple groups (having no non-trivial normal subgroups of their own), and $A_5$ is the smallest non-abelian simple group.

The symmetric groups have an intrinsic interest as permutation groups and can be studied in relation to the objects they permute (a basic form of group action).

If you are studying the symmetries of an object and you find an odd permutation, the even permutations will generally make up half the permutations you are studying and understanding what feature(s) of the object are preserved by even permutations but not odd ones may well help you to understand the object.

Note also that every group can be realised as a subgroup of a symmetric group (where the objects permuted are the elements of the group and the action is multiplication).