Products and Inverses in Permutation Groups

Permutation groups let us describe reordering in a precise mathematical way. students, if you have ever shuffled cards, mixed up the order of players in a lineup, or rearranged letters in a word, you have already seen the basic idea behind permutations 🎲. In this lesson, you will learn how to combine permutations using products, how to find inverses, and why these ideas matter in the study of abstract algebra.

What You Will Learn

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

explain what the product of two permutations means,
apply the rule for composing permutations correctly,
find the inverse of a permutation,
connect products and inverses to the structure of permutation groups,
use examples to verify your answers.

The key idea is simple but powerful: permutations are not just single rearrangements. They can be combined, and every permutation can be undone.

Products of Permutations

A permutation is a rearrangement of a set of objects. In abstract algebra, we often write permutations using cycle notation. For example, the cycle $(1\ 3\ 5)$ means that $1$ goes to $3$, $3$ goes to $5$, and $5$ goes back to $1$.

The product of two permutations means doing one permutation after the other. This is also called composition. If we write $\sigma\tau$, the standard convention is that $\tau$ is applied first, and then $\sigma$ is applied second. So the product $\sigma\tau$ means “do $\tau$, then do $\sigma$” ⏩.

This order matters a lot. In many cases,

$$\sigma\tau \neq \tau\sigma.$$

That means permutation multiplication is usually not commutative.

How to Compute a Product

To find the product of two permutations, follow each element through both rearrangements.

Suppose

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

and

$$\tau = (2\ 3).$$

To compute $\sigma\tau$, we apply $\tau$ first, then $\sigma$.

Let’s track each number:

$1$: $\tau$ leaves $1$ fixed, then $\sigma$ sends $1$ to $2$, so $1 \mapsto 2$
$2$: $\tau$ sends $2$ to $3$, then $\sigma$ sends $3$ to $1$, so $2 \mapsto 1$
$3$: $\tau$ sends $3$ to $2$, then $\sigma$ sends $2$ to $3$, so $3 \mapsto 3$

So the product is

$$\sigma\tau = (1\ 2).$$

Notice that $3$ ends up fixed, so it does not need to appear in the cycle notation.

Real-World Connection

Think of a classroom seating chart. One permutation might swap two students, while another might rotate three students to different seats. If you apply both rearrangements one after the other, the final result depends on the order. That is why the order of multiplication matters in permutation groups 🪑.

Cycle Notation and Products

Cycle notation makes it easier to understand products because it shows exactly where each element goes. However, cycles can be tricky when they overlap.

If two cycles involve disjoint sets of elements, they commute. For example,

$$ (1\ 2)(3\ 4) = (3\ 4)(1\ 2). $$

Why? Because the first cycle affects only $1$ and $2$, while the second affects only $3$ and $4$. Since they do not interfere, the order does not matter.

But if cycles share elements, the order usually changes the result. For example,

$$ (1\ 2)(2\ 3) \neq (2\ 3)(1\ 2). $$

This is one reason permutation groups are such an important example in abstract algebra: they show that not all groups behave the same way.

Example with a Full Calculation

Let

$$\alpha = (1\ 2\ 3\ 4)$$

and

$$\beta = (1\ 4).$$

Find $\alpha\beta$.

Again, apply $\beta$ first, then $\alpha$.

$1$: $\beta$ sends $1$ to $4$, and $\alpha$ sends $4$ to $1$, so $1 \mapsto 1$
$2$: $\beta$ leaves $2$ fixed, and $\alpha$ sends $2$ to $3$, so $2 \mapsto 3$
$3$: $\beta$ leaves $3$ fixed, and $\alpha$ sends $3$ to $4$, so $3 \mapsto 4$
$4$: $\beta$ sends $4$ to $1$, and $\alpha$ sends $1$ to $2$, so $4 \mapsto 2$

$$\alpha\beta = (2\ 3\ 4).$$

This example shows how products can simplify into a shorter cycle.

Inverses of Permutations

Every permutation has an inverse. The inverse is the rearrangement that “undoes” the original permutation 🔄. If a permutation sends $a$ to $b$, then its inverse sends $b$ back to $a$.

For a permutation $\sigma$, its inverse is written $\sigma^{-1}$. It satisfies

$$\sigma\sigma^{-1} = \sigma^{-1}\sigma = e,$$

where $e$ is the identity permutation, the permutation that leaves every element fixed.

Inverse of a Cycle

A useful rule is that the inverse of a cycle is found by reversing the cycle order.

For example,

$$ (1\ 2\ 3)^{-1} = (1\ 3\ 2). $$

Check it: $1 \mapsto 3 \mapsto 1$, $2 \mapsto 1 \mapsto 2$, and $3 \mapsto 2 \mapsto 3$.

In general, for a cycle

$$ (a_1\ a_2\ a_3\dots a_k), $$

its inverse is

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

Inverse of a Product

A very important rule is that the inverse of a product reverses the order:

$$ (\sigma\tau)^{-1} = \tau^{-1}\sigma^{-1}. $$

This is a common place where students make mistakes, so students, remember: reverse the order first, then invert each part.

Why does this work? Because

$$ (\sigma\tau)(\tau^{-1}\sigma^{-1}) = \sigma(\tau\tau^{-1})\sigma^{-1} = \sigma e\sigma^{-1} = \sigma\sigma^{-1} = e. $$

This proves that $\tau^{-1}\sigma^{-1}$ really is the inverse of $\sigma\tau$.

Worked Example: Product and Inverse Together

Suppose

$$\sigma = (1\ 2\ 4)$$

and

$$\tau = (2\ 4\ 3).$$

Step 1: Find the product $\sigma\tau$

Apply $\tau$ first, then $\sigma$.

$1$: $\tau$ leaves $1$ fixed, then $\sigma$ sends $1$ to $2$, so $1 \mapsto 2$
$2$: $\tau$ sends $2$ to $4$, then $\sigma$ sends $4$ to $1$, so $2 \mapsto 1$
$3$: $\tau$ sends $3$ to $2$, then $\sigma$ sends $2$ to $4$, so $3 \mapsto 4$
$4$: $\tau$ sends $4$ to $3$, then $\sigma$ leaves $3$ fixed, so $4 \mapsto 3$

Thus

$$\sigma\tau = (1\ 2)(3\ 4).$$

Step 2: Find the inverse

Since the product is a product of disjoint transpositions,

$$((1\ 2)(3\ 4))^{-1} = (3\ 4)^{-1}(1\ 2)^{-1} = (3\ 4)(1\ 2).$$

Because the cycles are disjoint, this is the same as

$$ (1\ 2)(3\ 4). $$

So this permutation is its own inverse.

This type of example shows how products and inverses work together in practice.

Why Products and Inverses Matter in Permutation Groups

The set of all permutations of a given set forms a group under composition. For $n$ objects, this group is called the symmetric group and is written $S_n$.

A group must satisfy four main properties:

closure,
associativity,
identity,
inverses.

Products and inverses are exactly the operations that help permutations satisfy these rules.

Closure means the product of two permutations is still a permutation.
Associativity means that if you have three permutations, the way you group them does not change the result:

$$ (\sigma\tau)\rho = \sigma(\tau\rho). $$

The identity permutation does nothing.
Every permutation has an inverse that undoes it.

These facts show why permutation groups are a central example in abstract algebra. They provide a concrete model for understanding group theory, even though the ideas are abstract.

Common Mistakes to Avoid

Here are a few mistakes students often make:

Reversing the order of multiplication incorrectly

Remember that $\sigma\tau$ means apply $\tau$ first.

Forgetting to reverse order when finding an inverse of a product

Always use

$$ (\sigma\tau)^{-1} = \tau^{-1}\sigma^{-1}. $$

Writing fixed points unnecessarily in cycle notation

If an element does not move, you do not need to include it.

Assuming all permutations commute

Usually,

$$ \sigma\tau \neq \tau\sigma. $$

Checking your work carefully with each element is the best way to avoid errors ✅.

Conclusion

Products and inverses are the heart of how permutation groups work. A product combines two rearrangements into one, while an inverse undoes a rearrangement. In cycle notation, the order of operations matters, and inverses require reversing that order. These ideas help explain why symmetric groups $S_n$ are such important examples in abstract algebra. students, if you can compute products and inverses confidently, you have a strong foundation for the rest of permutation groups and group theory.

Study Notes

A permutation is a rearrangement of objects.
The product of permutations means composition, and the rightmost permutation is applied first.
In general, permutation products are not commutative, so $\sigma\tau$ may not equal $\tau\sigma$.
Disjoint cycles commute, meaning their order does not affect the result.
The inverse of a permutation undoes it and satisfies $\sigma\sigma^{-1} = \sigma^{-1}\sigma = e$.
The inverse of a cycle is found by reversing the order of the elements.
The inverse of a product reverses the order: $$(\sigma\tau)^{-1} = \tau^{-1}\sigma^{-1}.$$
Permutation groups satisfy closure, associativity, identity, and inverses.
The symmetric group $S_n$ is the group of all permutations of $n$ objects.
Careful element-by-element tracking is the safest way to compute products and inverses.