ExampleConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck 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
$\operatorname{id}$	$[0,1,2]$	$0$	$+1$
$(0\,1)$	$[1,0,2]$	$1$	$-1$
$(0\,2)$	$[2,1,0]$	$3$	$-1$
$(1\,2)$	$[0,2,1]$	$1$	$-1$
$(0\,1\,2)$	$[1,2,0]$	$2$	$+1$
$(0\,2\,1)$	$[2,0,1]$	$2$	$+1$

Thus $A_3=\{\operatorname{id},(0\,1\,2),(0\,2\,1)\}$ .

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 $\operatorname{sgn}(\sigma)=(-1)^{\operatorname{inv}(\sigma)}$ , and even and odd permutations, The alternating group $A_n=\ker(\operatorname{sgn})$ of even permutations).

[L2]

A three-element set has $3!=6$ permutations (A finite set $A$ with $\lvert A\rvert = n$ 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 · next 3 levels

Direct dependencies and their dependencies through the next three levels: 66 results over 16 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click 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 S3S_3S3​: one-line form, cycle structure, inversions, and sign