ExampleConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-02

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Conjugation by $(1\,2)$ in $\operatorname{Sym}(\{1,2,3\})$ exchanges the transpositions $(1\,3)$ and $(2\,3)$

Example

Conjugation by $(1\,2)$ in $\operatorname{Sym}(\{1,2,3\})$ exchanges the transpositions $(1\,3)$ and $(2\,3)$ .

Facts & Assumptions

Given: The symmetric group on $\{1,2,3\}$ , with composition acting right-to-left.

[L1]

Conjugation is an automorphism (Conjugation $x\mapsto gxg^{-1}$ is an automorphism).

[L2]

Inner automorphisms are conjugations (Inner automorphisms and $\operatorname{Inn}(G)$ ).

[L3]

Elements of $\operatorname{Sym}(\{1,2,3\})$ are bijections composed right-to-left (The symmetric group $\operatorname{Sym}(X)$ : the bijections of a set $X$ under composition).

[L4]

These bijections form a group ( $\operatorname{Sym}(X)$ is a group under composition, and it is non-abelian whenever $X$ has at least three distinct elements).

Verification

technique · direct

1.1

Direct evaluation on $1,2,3$ gives $(1\,2)(1\,3)(1\,2)=(2\,3)$ .

L1L2L3L4givenalgebra

2.1

Since $(1\,2)^{-1}=(1\,2)$ , the same computation with the roles reversed gives $(1\,2)(2\,3)(1\,2)=(1\,3)$ .

step 1.1L1L2L3L4givenalgebra

3.1

Thus this inner automorphism exchanges the two stated transpositions.

step 2.1∎

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: 20 results over 14 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

Milne, Group Theory, Automorphisms of Groups (standard reference, not scraped)

Conjugation by (1 2)(1\,2)(12) in Sym⁡({1,2,3})\operatorname{Sym}(\{1,2,3\})Sym({1,2,3}) exchanges the transpositions (1 3)(1\,3)(13) and (2 3)(2\,3)(23)