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Understanding Permutations and the Symmetric Group S3

Introduction

Have you ever rearranged the letters in a word, shuffled a deck of cards, or mixed up your schedule? If so, you’ve worked with permutations — rearrangements of a set of objects. In this post, we'll explore permutations in a mathematical way using functions, and learn how to combine them using composition. Then, we’ll look at the symmetric group S3S_3 — the set of all possible ways to rearrange 3 things — and build its Cayley table, which shows how these rearrangements interact.

No worries if you’ve never heard of abstract algebra — we’ll break it all down!

📦 What Is a Permutation?

A permutation is a way of rearranging a set of objects. Suppose we have the set {1,2,3}\{1, 2, 3\}. One permutation might send:

  • 1 to 2,
  • 2 to 3,
  • 3 to 1.

We write that as:

(123231)\left(\begin{matrix} 1 & 2 & 3 \\ 2 & 3 & 1 \end{matrix}\right)

This is just a fancy way to show where each number goes — a kind of function that scrambles the set.

🔁 Composing Functions

Function composition means applying one function, then another. For example, if:

  • Function ff sends 1 to 2,
  • and function gg sends 2 to 3,

then the composition g∘fg \circ f sends 1 to 3 (because 1 goes to 2 via ff, then 2 goes to 3 via gg).

We'll use this idea to combine permutations.

🔢 The Elements of S3S_3

The group S3S_3 has all the permutations of 3 elements. There are 6 in total:

Label Two-Line Notation Cycle Notation
σ1\sigma_1 (123123)\left(\begin{smallmatrix}1 & 2 & 3\\ 1 & 2 & 3\end{smallmatrix}\right) identity
σ2\sigma_2 (123213)\left(\begin{smallmatrix}1 & 2 & 3\\ 2 & 1 & 3\end{smallmatrix}\right) (12)
σ3\sigma_3 (123321)\left(\begin{smallmatrix}1 & 2 & 3\\ 3 & 2 & 1\end{smallmatrix}\right) (13)
σ4\sigma_4 (123132)\left(\begin{smallmatrix}1 & 2 & 3\\ 1 & 3 & 2\end{smallmatrix}\right) (23)
σ5\sigma_5 (123231)\left(\begin{smallmatrix}1 & 2 & 3\\ 2 & 3 & 1\end{smallmatrix}\right) (123)
σ6\sigma_6 (123312)\left(\begin{smallmatrix}1 & 2 & 3\\ 3 & 1 & 2\end{smallmatrix}\right) (132)

Each one is a unique way to reorder the numbers 1, 2, and 3.

🧠 How to Compose Two Permutations

Let’s say we want to compute σ4∘σ6\sigma_4 \circ \sigma_6. This means: first apply σ6\sigma_6, then apply σ4\sigma_4 to the result.

Here are the steps:

  • σ6(1)=3\sigma_6(1) = 3, then σ4(3)=2\sigma_4(3) = 2, so σ4∘σ6(1)=2\sigma_4 \circ \sigma_6(1) = 2
  • σ6(2)=1\sigma_6(2) = 1, then σ4(1)=1\sigma_4(1) = 1, so σ4∘σ6(2)=1\sigma_4 \circ \sigma_6(2) = 1
  • σ6(3)=2\sigma_6(3) = 2, then σ4(2)=3\sigma_4(2) = 3, so σ4∘σ6(3)=3\sigma_4 \circ \sigma_6(3) = 3

Putting it together:

σ4∘σ6=(123213)=σ2\sigma_4 \circ \sigma_6 = \left(\begin{matrix} 1 & 2 & 3 \\ 2 & 1 & 3 \end{matrix}\right) = \sigma_2

Nice!

🧮 Full Cayley Table of S3S_3

This table shows the result of composing any two permutations from S3S_3. Rows are the first function, columns are the second. You apply the one from the row, then the one from the column.

∘\circ σ1\sigma_1 σ2\sigma_2 σ3\sigma_3 σ4\sigma_4 σ5\sigma_5 σ6\sigma_6
σ1\sigma_1 σ1\sigma_1 σ2\sigma_2 σ3\sigma_3 σ4\sigma_4 σ5\sigma_5 σ6\sigma_6
σ2\sigma_2 σ2\sigma_2 σ1\sigma_1 σ6\sigma_6 σ5\sigma_5 σ4\sigma_4 σ3\sigma_3
σ3\sigma_3 σ3\sigma_3 σ5\sigma_5 σ1\sigma_1 σ6\sigma_6 σ2\sigma_2 σ4\sigma_4
σ4\sigma_4 σ4\sigma_4 σ6\sigma_6 σ5\sigma_5 σ1\sigma_1 σ3\sigma_3 σ2\sigma_2
σ5\sigma_5 σ5\sigma_5 σ3\sigma_3 σ2\sigma_2 σ4\sigma_4 σ6\sigma_6 σ1\sigma_1
σ6\sigma_6 σ6\sigma_6 σ4\sigma_4 σ3\sigma_3 σ2\sigma_2 σ1\sigma_1 σ5\sigma_5

This shows every combination — all 36 possible compositions.

🔄 Comparing Notation

Let’s compare one element in both notations:

σ5=(123231)=(123)\sigma_5 = \left(\begin{matrix} 1 & 2 & 3 \\ 2 & 3 & 1 \end{matrix}\right) = (123)

This tells us:

  • 1 → 2
  • 2 → 3
  • 3 → 1

Cycle notation shows the movement in one loop: 1 to 2 to 3 to 1. It's shorter and more intuitive once you get used to it.

🎉 Conclusion

You just learned:

  • What permutations are
  • How to write and compose them
  • The structure of the symmetric group S3S_3
  • How to compute every element of its Cayley table
  • How to compare full notation and cycle notation

This is a first peek into group theory, a branch of abstract algebra with deep applications in math, science, and even art and cryptography.

💬 Want to Try It?

Try computing some entries from the table yourself. For example:

  • What is σ5∘σ3\sigma_5 \circ \sigma_3?
  • Can you find an element that, when composed with σ4\sigma_4, gives the identity?

Drop your answers or questions in the comments!

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