If you rearrange some objects, and then rearrange the result, the overall effect is itself a rearrangement. This is composition — doing one permutation, then another.

How composition works

Suppose $\sigma$ sends $1 \to 2, 2 \to 3, 3 \to 1$, and $\tau$ swaps 1 and 2 (leaving 3 alone).

To compute $\tau \circ \sigma$ (“do $\sigma$ first, then $\tau$”), trace each element through both steps:

• $1 \xrightarrow{\sigma} 2 \xrightarrow{\tau} 1$, so $1 \to 1$
• $2 \xrightarrow{\sigma} 3 \xrightarrow{\tau} 3$, so $2 \to 3$
• $3 \xrightarrow{\sigma} 1 \xrightarrow{\tau} 2$, so $3 \to 2$

The result swaps 2 and 3, leaving 1 in place. In two-line notation:

$$\begin{pmatrix} 1 & 2 & 3 \\ 2 & 3 & 1 \end{pmatrix} \text{ then } \begin{pmatrix} 1 & 2 & 3 \\ 2 & 1 & 3 \end{pmatrix} = \begin{pmatrix} 1 & 2 & 3 \\ 1 & 3 & 2 \end{pmatrix}$$

The recipe: for each element in the top row, read its image from the bottom row of $\sigma$, then look up that element in $\tau$.

Two key observations

The result is always a permutation

No matter which two permutations you compose, you get another permutation. The result still sends every element somewhere, and no two elements collide. This property is called closure — the set of permutations is closed under composition.

Order matters

Try composing the same two permutations in the opposite order. Usually, you get a different result!

Swapping 1 and 2 then rotating gives a different outcome from rotating then swapping. Permutation composition is not commutative — the order in which you apply the operations matters. (This is why we’re careful about “$\sigma$ first, then $\tau$.”)

Connecting to group theory

These two observations — closure and non-commutativity — are exactly what makes permutations interesting as algebraic structures. When we study groups, we’ll see that the full set of permutations satisfies all four group axioms, and the non-commutativity tells us these groups are not abelian (except for very small sets).