False statementConstruction: Literature-sourcedVerification: Literature-sourcedprecheck passaudited 2026-08-13

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FALSE: two even permutations of the same cycle type are always conjugate in $A_{n}$

Statement refuted

Two elements of $A_{n}$ with the same cycle type must be conjugate in $A_{n}$ .

Facts & Assumptions

Given: The cycles $(123)$ and $(132)$ in $A_{4}$ .

[F1]

A $k$ -cycle has sign $(- 1)^{k - 1}$ (A $k$ -cycle has sign $(- 1)^{k - 1}$ , and $sgn (σ) = (- 1)^{n - c (σ)}$ when fixed points are counted as cycles), and $A_{n}$ is the kernel of sign (The alternating group $A_{n} = \ker (sgn)$ of even permutations).

[F2]

Conjugation relabels cycle entries (Conjugating a cycle relabels each entry: $g (a_{1} \dots a_{k}) g^{- 1} = (g (a_{1}) \dots g (a_{k}))$ ).

[F3]

For $n \geq 2$ and $σ \in A_{n}$ , the $S_{n}$ -class of $σ$ splits into two $A_{n}$ -classes of equal size exactly when all cycle lengths in its decomposition, including $1$ -cycles for fixed points, are odd and no two are equal (For $n \geq 2$ , an $S_{n}$ -class of an even permutation splits in $A_{n}$ exactly when all cycle lengths, including $1$ -cycles, are odd and distinct).

Counterexample

technique · counterexample

1.1

In $A_{4}$ , the cycles $(123)$ and $(132)$ have sign $(- 1)^{2} = + 1$ by [F1] and have the same cycle type $(3, 1)$ .

F1algebra

2.1

The lengths $3$ and $1$ are odd and distinct, so [F3] says that the $S_{4}$ -class of $3$ -cycles splits into two $A_{4}$ -classes.

F3step 1.1

2.2

More explicitly, [F2] shows that conjugating $(123)$ to $(132)$ reverses the cyclic order on ${1, 2, 3}$ while fixing the remaining point; every such relabeling is odd. Hence no element of $A_{4}$ performs it.

F2step 1.1algebra

3.1

Thus the two displayed even permutations have the same cycle type but are not conjugate in $A_{4}$ .

step 1.1step 2.2∎

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

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

FALSE: two even permutations of the same cycle type are always conjugate in AnAn​