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Conjugating $(1 4) (2 5 3)$ by an explicit permutation in $S_{5}$

Example

Let $σ = (1 4) (2 5 3)$ and $g = (1 2 3 4 5)$ in $S_{5}$ . Then $g σ g^{- 1} = (2 5) (3 1 4) .$

Facts & Assumptions

Given: $σ = (1 4) (2 5 3)$ and $g = (1 2 3 4 5)$ in $S_{5}$ .

[F1]

Conjugating a cycle relabels each entry by the conjugating permutation (Conjugating a cycle relabels each entry: $g (a_{1} \dots a_{k}) g^{- 1} = (g (a_{1}) \dots g (a_{k}))$ ).

[F2]

Permutations in $S_{n}$ are conjugate exactly when they have the same cycle type (Two elements of $S_{n}$ are conjugate if and only if they have the same cycle type).

Verification

technique · direct

1.1

The relabellings are $(g (1) g (4)) = (2 5)$ and $(g (2) g (5) g (3)) = (3 1 4)$ .

algebra

2.1

Apply [F1] to the two disjoint factors to obtain the displayed conjugate.

F1step 1.1

3.1

The result has one $2$ -cycle and one $3$ -cycle, hence the same cycle type as $σ$ , as [F2] requires; conjugating back by $g^{- 1}$ recovers $σ$ .

F2step 2.1algebra∎

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: 15 results over 7 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

K. Conrad, Conjugacy Classes (standard reference, not scraped)

Conjugating (1 4)(2 5 3)(14)(253) by an explicit permutation in S5S5​