Two-line notation works, but it’s verbose — especially for larger permutations. Cycle notation is a compact way to describe a permutation by recording which elements move in loops.

Reading a cycle

The cycle $(1\ 2\ 3)$ means: 1 goes to 2, 2 goes to 3, 3 goes back to 1. Follow the cycle like a loop:

$$1 \to 2 \to 3 \to 1$$

The cycle $(1\ 2)$ means: 1 goes to 2, 2 goes back to 1 — a transposition (a swap).

Elements not mentioned in any cycle stay in place. So for a permutation of ${1, 2, 3, 4}$, the notation $(1\ 2)$ means “swap 1 and 2, leave 3 and 4 alone.”

The identity

The identity permutation — the one that moves nothing — is written $e$ (or sometimes $(\ )$ or $\text{id}$). No cycles needed because nothing moves.

Disjoint cycles

A permutation might have several independent loops. We write them side by side:

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

This means: swap 1 with 2 AND swap 3 with 4. These cycles are disjoint — they don’t share any elements, so they don’t interfere with each other.

Every permutation can be written uniquely (up to cycle order) as a product of disjoint cycles.

Converting from two-line notation

Start at 1. Follow where it goes. Keep following until you return to 1 — that’s your first cycle. Then start with the smallest element you haven’t visited yet.

For example:

$$\begin{pmatrix} 1 & 2 & 3 & 4 \\ 3 & 4 & 1 & 2 \end{pmatrix}$$

Start at 1: $1 \to 3 \to 1$. First cycle: $(1\ 3)$. Next unvisited: 2: $2 \to 4 \to 2$. Second cycle: $(2\ 4)$. Result: $(1\ 3)(2\ 4)$.

Why cycle notation matters

Cycle notation isn’t just shorter — it reveals the structure of a permutation at a glance:

• Cycle type: $(1\ 2\ 3)(4\ 5)$ is a “3-cycle times a 2-cycle.” The cycle type tells you a lot about the permutation’s algebraic properties.
• Order: The order of a permutation (how many times you need to apply it to get back to the identity) equals the least common multiple of its cycle lengths. $(1\ 2\ 3)(4\ 5)$ has order $\text{lcm}(3, 2) = 6$.
• Parity: A permutation is even or odd depending on whether it decomposes into an even or odd number of transpositions. This distinction is fundamental to Galois theory — the alternating group $A_n$ consists of exactly the even permutations.