A permutation is a shuffle
A permutation of a set is a rearrangement of it — formally, a one-to-one function from the set onto itself. Think of three chairs labelled 1, 2, 3 and three people sitting in them. A permutation is any rule that reseats everyone so that each chair still holds exactly one person. There are 3! = 6 ways to do this for three objects, and n! ways for n — the factorial counts them.
A compact way to write a permutation is cycle notation. The cycle (1 2 3) means “1 goes to 2, 2 goes to 3, 3 goes back to 1” — a single loop. The cycle (1 2) means “swap 1 and 2, leave 3 alone” — a transposition. Anything left out of every cycle stays fixed. The do-nothing permutation, which fixes everyone, is the [[identity-element-of-a-group|identity]], written e.
Composing shuffles, and why order matters
The operation on permutations is composition: do one shuffle, then another. We'll read right-to-left, the way functions compose — in σ ∘ τ, apply τ first, then σ. Composition of permutations is always associative, the identity does nothing, and every permutation has an inverse, so the set of all permutations of n objects is a group. It is the [[symmetric-group|symmetric group]] Sₙ, and it has n! elements.
Work in S₃ on {1,2,3}. Let σ = (1 2 3) and τ = (1 2).
Compose right-to-left: apply the right one first.
σ ∘ τ (do τ, then σ):
1 --τ--> 2 --σ--> 3 so 1 → 3
2 --τ--> 1 --σ--> 2 so 2 → 2 (fixed)
3 --τ--> 3 --σ--> 1 so 3 → 1
result: 1→3, 3→1, 2 fixed = (1 3)
τ ∘ σ (do σ, then τ):
1 --σ--> 2 --τ--> 1 so 1 → 1 (fixed)
2 --σ--> 3 --τ--> 3 so 2 → 3
3 --σ--> 1 --τ--> 2 so 3 → 2
result: 2→3, 3→2, 1 fixed = (2 3)
σ ∘ τ = (1 3) but τ ∘ σ = (2 3)
(1 3) ≠ (2 3) → composition does NOT commute.That asymmetry is the headline. For n ≥ 3, Sₙ is not abelian — the order in which you shuffle changes the outcome. These are the most natural non-commutative groups you will meet, and they are the reason commutativity was singled out as optional back in the first guide. The order of a single cycle equals its length: (1 2 3) has order 3 (three turns return home), while a transposition like (1 2) has order 2.
Why symmetric groups sit at the center
Symmetric groups aren't just one example among many — they're universal. Cayley's theorem states that *every* finite group is (a copy of) a subgroup of some symmetric group. The reason is intuitive: each element of a group, acting by combination, simply reshuffles the group's own elements, and a reshuffle is a permutation. So the abstract notion of “group” and the concrete notion of “collection of shuffles” are, at bottom, the same idea.