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

Example

Conjugation by $(1 2)$ in $Sym ({1, 2, 3})$ 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 $x \mapsto g x g^{- 1}$ is an automorphism).

[L2]

Inner automorphisms are conjugations (Inner automorphisms and $Inn (G)$ ).

[L3]

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

[L4]

These bijections form a group ( $Sym (X)$ 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 · two levels

13 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

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

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