Orders of products in symmetric groups

DefinitionKohl's question on orders of products in symmetric groups#✓fully proved

For every natural number \(n\) and all natural numbers \(a, b, c\) with \(1 < a \le n - 2\), \(1 < b \le n - 2\) and \(1 < c \le n - 2\), the symmetric group \(S_n\) contains permutations \(x\) and \(y\) such that \(x\) has order \(a\), \(y\) has order \(b\) and \(xy\) has order \(c\).

def symmetricGroupHasProductWithOrders : Prop :=
  ∀ n a b c : ℕ, 1 < a → 1 < b → 1 < c → a ≤ n - 2 → b ≤ n - 2 → c ≤ n - 2 →
    ∃ x y : Equiv.Perm (Fin n), orderOf x = a ∧ orderOf y = b ∧ orderOf (x * y) = c

Used by 1

Used byPermutations of prescribed orders with product of prescribed order

TheoremPermutations of prescribed orders with product of prescribed order#●ready to prove

For every natural number \(n\) and all natural numbers \(a, b, c\) with \(1 < a \le n - 2\), \(1 < b \le n - 2\) and \(1 < c \le n - 2\), the symmetric group \(S_n\) contains permutations \(x\) and \(y\) of orders \(a\) and \(b\) whose product \(xy\) has order \(c\).

public theorem_wanted symmetricGroupHasProductWithOrders_holds : symmetricGroupHasProductWithOrders

Sources

• E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Archive of Solved Problems, Problem 18.49 (S. Kohl); resolution: G. A. Miller, Amer. J. Math. 22 (1900), no. 2, 185-190; J. König, European J. Combin. 57 (2016), 50-56, Kourovka Archive, Problem 18.49: Yes, it is (Miller); another solution by König, https://arxiv.org/abs/1401.0300v49.

Uses 1

Statement usesKohl's question on orders of products in symmetric groups

1 fully proved1 ready to prove

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