ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-08-11

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The six elements of $S_{3}$ : one-line form, cycle structure, inversions, and sign

Example

The elements of $S_{3}$ have the following one-line forms, cycle decompositions, inversion numbers, and signs:

permutation	one-line form	inversion number	sign
$id$	$[0, 1, 2]$	$0$	$+ 1$
$(0 1)$	$[1, 0, 2]$	$1$	$- 1$
$(0 2)$	$[2, 1, 0]$	$3$	$- 1$
$(1 2)$	$[0, 2, 1]$	$1$	$- 1$
$(0 1 2)$	$[1, 2, 0]$	$2$	$+ 1$
$(0 2 1)$	$[2, 0, 1]$	$2$	$+ 1$

Thus $A_{3} = {id, (0 1 2), (0 2 1)}$ .

Facts & Assumptions

Given: The symmetric group $S_{3}$ acting on $3 = {0, 1, 2}$ .

[L1]

The inversion number counts decreasing pairs in one-line notation, sign is $(- 1)$ to that number, and $A_{3}$ is the set of even permutations (Inversions, inversion number, the sign $sgn (σ) = (- 1)^{inv (σ)}$ , and even and odd permutations, The alternating group $A_{n} = \ker (sgn)$ of even permutations).

[L2]

A three-element set has $3! = 6$ permutations (A finite set $A$ with $∣ A ∣ = n$ has exactly $n!$ bijections onto itself, and $n!$ bijections onto any set of the same cardinality).

Verification

technique · direct

1.1

By [L2] there are $3! = 6$ permutations. Listing the three transpositions, the two three-cycles, and the identity gives six distinct maps. Counting decreasing pairs in each displayed one-line form gives respectively $0, 1, 3, 1, 2, 2$ , and [L1] gives the signs shown.

givenL1L2

2.1

The rows with sign $+ 1$ are exactly the identity and the two three-cycles, so the displayed set is precisely $A_{3}$ .

step 1.1L1∎

Depends on

Used by

Nothing in the library uses this result yet.

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 (standard reference, not scraped)
Stanford Math 51H, Permutations (standard reference, not scraped)

The six elements of S3S3​: one-line form, cycle structure, inversions, and sign