Alternating Groups Overview

students, imagine a group of friends lining up for a class photo 📸. Some rearrangements of the line are simple swaps, while others involve a more complex reshuffling. In abstract algebra, these rearrangements are called permutations. Today’s lesson looks at a special and very important set of permutations: the alternating groups.

By the end of this lesson, you should be able to:

Explain what alternating groups are and why they matter.
Tell whether a permutation is even or odd.
Understand how alternating groups fit inside symmetric groups.
Use examples to work with alternating groups in permutation problems.
Recognize why alternating groups are a key part of the study of permutation groups.

What Is an Alternating Group?

An alternating group is the set of all even permutations of a finite set. If the set has $n$ elements, the alternating group is written as $A_n$.

To understand this, first remember that the symmetric group $S_n$ is the group of all permutations of $n$ objects. Every permutation in $S_n$ can be written as a product of transpositions, and a transposition is a swap of just two elements. For example, the permutation $(1\ 2)$ swaps $1$ and $2$.

A permutation is called even if it can be written as a product of an even number of transpositions. It is called odd if it can be written as a product of an odd number of transpositions.

So $A_n$ is the collection of all even permutations in $S_n$.

For example, in $S_3$ the permutation $(1\ 2\ 3)$ is even because it can be written as

$$(1\ 2\ 3) = (1\ 3)(1\ 2)$$

This uses two transpositions, so it is even. On the other hand, the transposition $(1\ 2)$ is odd because it is a single transposition.

This idea matters because it splits the symmetric group into two parts: even permutations and odd permutations. Together they make up all of $S_n$.

Even and Odd Permutations

students, one of the most important facts in this topic is that a permutation is either even or odd, and it cannot be both. This is not just a definition trick. There is a real theorem behind it: the parity of a permutation is well defined.

That means if a permutation can be written as a product of transpositions in more than one way, every such way has either always an even number of transpositions or always an odd number of transpositions. You never get one description with even length and another with odd length for the same permutation.

This makes it possible to classify each permutation by parity.

Let’s look at a simple example in $S_4$.

Consider the cycle $(1\ 2\ 3)$. It can be written as

$$(1\ 2\ 3) = (1\ 3)(1\ 2)$$

That is two transpositions, so it is even. Now consider the transposition $(2\ 4)$. It is already a single transposition, so it is odd.

A useful pattern is this: a cycle of length $k$ can be written as a product of $k-1$ transpositions:

$$(a_1\ a_2\ \dots\ a_k) = (a_1\ a_k)(a_1\ a_{k-1})\cdots(a_1\ a_2)$$

So the parity of a cycle depends on whether $k-1$ is even or odd. For example:

A $2$-cycle is odd.
A $3$-cycle is even.
A $4$-cycle is odd.
A $5$-cycle is even.

This is a fast way to identify many elements of $A_n$.

Why Alternating Groups Are a Group

It is not enough that $A_n$ is a set of permutations. It must also be a group. Fortunately, it is.

The key reason is that the product of two even permutations is even, and the inverse of an even permutation is also even.

Here is the intuition:

If one permutation uses an even number of transpositions and another also uses an even number, then combining them gives an even total parity.
If a permutation is even, its inverse has the same parity because reversing a product of transpositions does not change whether the number is even or odd.

So $A_n$ satisfies the group properties under composition.

This makes $A_n$ a subgroup of $S_n$.

In fact, $A_n$ is a very important subgroup because it has exactly half the size of $S_n$.

Since there are $n!$ permutations in $S_n$, there are

$$|A_n| = \frac{n!}{2}$$

even permutations in $A_n$.

This is a strong and useful result. It shows that half of all permutations are even and half are odd.

Examples in Small Cases

Let’s explore the first few alternating groups.

The group $A_1$

There is only one permutation of a one-element set: the identity. The identity is even, so

$$A_1 = S_1$$

The group $A_2$

The group $S_2$ has two elements: the identity and $(1\ 2)$.

The identity is even.
$(1\ 2)$ is odd.

$$A_2 = \{e\}$$

The group $A_3$

The group $S_3$ has $6$ elements. The even ones are the identity and the two $3$-cycles:

$e$
$(1\ 2\ 3)$
$(1\ 3\ 2)$

Thus,

$$A_3 = \{e, (1\ 2\ 3), (1\ 3\ 2)\}$$

This group has $3$ elements and is cyclic. In fact, it is isomorphic to $C_3$.

The group $A_4$

The group $A_4$ is especially important. It has $12$ elements, since

$$|A_4| = \frac{4!}{2} = 12$$

Its elements include:

the identity,
the $3$-cycles,
and the products of two disjoint transpositions, such as $(1\ 2)(3\ 4)$.

Notice that $(1\ 2)(3\ 4)$ is even because it is a product of two transpositions.

The group $A_4$ is often studied because it is the smallest alternating group that is not cyclic and not simple in a trivial way. It also appears in many structural questions in group theory.

Alternating Groups and Simplicity

One major reason alternating groups are important is that many of them are simple groups.

A group is simple if it has no normal subgroups other than the identity subgroup and the whole group.

The alternating groups $A_n$ are simple for all $n \ge 5$.

This is a famous theorem in group theory and one of the reasons alternating groups play such a central role. Simple groups are like the “building blocks” of group theory, similar to how prime numbers are the building blocks of arithmetic 🔍

For example:

$A_5$ is simple.
$A_6$, $A_7$, and all larger alternating groups are also simple.

This fact has deep consequences in mathematics, especially in understanding how groups can be broken into smaller pieces.

You do not need to prove simplicity at this stage, but you should know the statement and why it is important.

How to Tell Whether a Permutation Is in $A_n$

To decide whether a permutation belongs to an alternating group, check whether it is even.

Here are some quick steps students can use:

Write the permutation as a product of cycles.
Break each cycle into transpositions if needed.
Count the total number of transpositions.
If the total is even, the permutation is in $A_n$.
If the total is odd, it is not in $A_n$.

Example:

Is $(1\ 2\ 3)(4\ 5)$ in $A_5$?

$(1\ 2\ 3)$ is a $3$-cycle, so it is even.
$(4\ 5)$ is a transposition, so it is odd.
Even $\times$ odd gives odd.

So $(1\ 2\ 3)(4\ 5)$ is not in $A_5$.

Now try $(1\ 2\ 3)(4\ 5)(6\ 7)$ in $S_7$.

$(1\ 2\ 3)$ is even.
$(4\ 5)$ is odd.
$(6\ 7)$ is odd.
Odd $\times$ odd gives even.

So the whole permutation is even, and it belongs to $A_7$.

Connection to the Broader Topic of Permutation Groups

Alternating groups are a special kind of permutation group, so they fit directly into the bigger study of permutations.

Here is the big picture:

Symmetric groups $S_n$ contain all permutations of $n$ objects.
Alternating groups $A_n$ contain exactly the even permutations.
Cycle notation helps describe elements in both groups.
Products and inverses work the same way because these groups are defined by composition.

So when you study alternating groups, you are building on everything already learned about cycle notation, products of permutations, and inverses.

This makes alternating groups a natural next step after learning the basics of permutation groups.

In real-world terms, permutation groups can model rearrangements of objects, and alternating groups describe the rearrangements that can be built from an even number of swaps. That makes them a refined and highly structured subset of all possible rearrangements.

Conclusion

Alternating groups are the groups of even permutations, written as $A_n$. They sit inside the symmetric groups $S_n$ and contain exactly half of the permutations of $n$ objects. A permutation is even if it can be written as a product of an even number of transpositions, and this parity is always well defined.

students, you should now be able to identify alternating groups, check whether a permutation is even or odd, and understand why these groups are important in the broader study of permutation groups. Alternating groups are not just a side topic—they are a central structure in abstract algebra, especially because many of them are simple for $n \ge 5$.

Study Notes

$S_n$ is the symmetric group of all permutations of $n$ objects.
$A_n$ is the alternating group of all even permutations in $S_n$.
A permutation is even if it can be written as a product of an even number of transpositions.
A permutation is odd if it can be written as a product of an odd number of transpositions.
The parity of a permutation is well defined.
A cycle of length $k$ can be written as a product of $k-1$ transpositions.
Therefore, $3$-cycles are even and transpositions are odd.
The size of $A_n$ is $\frac{n!}{2}$.
$A_3 = \{e, (1\ 2\ 3), (1\ 3\ 2)\}$.
$A_4$ has $12$ elements and includes products like $(1\ 2)(3\ 4)$.
For all $n \ge 5$, the alternating group $A_n$ is simple.
Alternating groups are an essential part of permutation groups and help connect cycle notation, products, and inverses.