TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-11

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Every finite permutation is a product of transpositions, so the transpositions generate $S_{n}$

Statement

Every permutation of a finite set is a product of transpositions. Consequently, for every natural $n$ , the transpositions in $S_{n} = Sym (n)$ generate $S_{n}$ . The identity, including the only permutations in $S_{0}$ and $S_{1}$ , is represented by the empty product.

Facts & Assumptions

Given: A finite set $X$ and a permutation $σ \in Sym (X)$ , with the right-hand factor in a product acting first.

[L1]

Every finite permutation is a product of pairwise disjoint cycles, with the identity represented by the empty product (Every permutation of a finite set is a product of pairwise disjoint cycles, uniquely up to reordering and cyclic rotation).

[L2]

The subgroup generated by a subset is the smallest subgroup containing that subset (The subgroup $⟨ S ⟩$ generated by a subset, the cyclic subgroup $⟨ g ⟩$ , and cyclic groups).

Proof

technique · direct

1.1

For $k \geq 2$ , pointwise evaluation gives $(a_{0} a_{1} \dots a_{k - 1}) = (a_{0} a_{k - 1}) \dots (a_{0} a_{2}) (a_{0} a_{1})$ : the rightmost factor sends $a_{0}$ to $a_{1}$ , each $a_{i}$ to $a_{i + 1}$ , and the leftmost factor sends $a_{k - 1}$ back to $a_{0}$ , while all other points are fixed.

givenL1

2.1

Replace each cycle in the decomposition supplied by [L1] with the factorisation in step 1.1 and concatenate the resulting finite lists. This expresses $σ$ as a product of transpositions.

step 1.1L1

3.1

If $σ$ is the identity, the decomposition and the resulting list are empty. Thus the conclusion includes $n = 0$ and $n = 1$ , and every element of $S_{n}$ lies in the smallest subgroup containing all transpositions, which by [L2] says that the transpositions generate $S_{n}$ .

step 2.1L1L2∎

Depends on

Used by

Dependency tree · two levels

15 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

T. W. Judson, Abstract Algebra: Theory and Applications, §5.1, Proposition 5.4 (standard reference, not scraped)
J. S. Milne, Group Theory, Corollary 4.27 (standard reference, not scraped)

Every finite permutation is a product of transpositions, so the transpositions generate SnSn​