Largest orbits of linear group actions

DefinitionKeller's question on two largest orbits and the dihedral group of order eight (Kourovka Problem 18.44)#✓fully proved

Keller's question on two largest orbits and the dihedral group of order eight (Kourovka Problem 18.44). The statement that for every finite non-abelian group \(G\), with derived subgroup \(G'\), and every finite faithful irreducible \(G\)-module \(V\) (a nonzero finite abelian group on which \(G\) acts faithfully by automorphisms with no \(G\)-invariant subgroups other than \(0\) and \(V\)), writing \(M = |G/G'|\): if every orbit of \(G\) on \(V\) has at most \(M\) elements and exactly two orbits have exactly \(M\) elements, then \(G\) is isomorphic to the dihedral group of order \(8\) and \(|V| = 9\).

def twoLargestOrbitsImpliesDihedralEight : Prop :=
  ∀ (G : Type) [Group G] [Finite G] (V : Type) [AddCommGroup V] [Finite V]
    [DistribMulAction G V], (∃ a b : G, a * b ≠ b * a) → FaithfulSMul G V → Nontrivial V →
    (∀ W : AddSubgroup V, (∀ g : G, ∀ w ∈ W, g • w ∈ W) → W = ⊥ ∨ W = ⊤) →
    (∀ v : V, Nat.card (MulAction.orbit G v) ≤ Nat.card (Abelianization G)) →
    {O : MulAction.orbitRel.Quotient G V |
      Nat.card O.orbit = Nat.card (Abelianization G)}.ncard = 2 →
    Nonempty (G ≃* DihedralGroup 4) ∧ Nat.card V = 9

Used by 1

Used byTwo largest orbits force the dihedral group of order eight (Kourovka 18.44)

TheoremTwo largest orbits force the dihedral group of order eight (Kourovka 18.44)#●ready to prove

Let \(G\) be a finite non-abelian group, let \(G'\) be its derived subgroup, and let \(V\) be a finite faithful irreducible \(G\)-module, that is, a nonzero finite abelian group on which \(G\) acts faithfully by automorphisms with no \(G\)-invariant subgroups other than \(0\) and \(V\). Write \(M = |G/G'|\). If every orbit of \(G\) on \(V\) has at most \(M\) elements and exactly two orbits have exactly \(M\) elements, then \(G\) is isomorphic to the dihedral group of order \(8\) and \(|V| = 9\).

Jones and Keller (2020), Theorem 3.1, prove this, showing moreover that \(V\) is the \(2\)-dimensional vector space \(V(2, 3)\) over the field with \(3\) elements; the Kourovka Notebook (21st issue) accepts it as answering 18.44.

public theorem_wanted twoLargestOrbitsImpliesDihedralEight_holds :
    twoLargestOrbitsImpliesDihedralEight

Sources

• E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Problem 18.44 (T. M. Keller); resolution: N. A. Jones and T. M. Keller, Orbit sizes and the dihedral group of order eight, Ann. Mat. Pura Appl. 199 (2020), 2323-2340, Theorem 3.1: G finite nonabelian, V finite faithful irreducible G-module, M = |G/G'| the largest orbit size with exactly two orbits of size M; then G is dihedral of order 8 and V = V(2, 3), https://doi.org/10.1007/s10231-020-00970-x.

Uses 1

Statement usesKeller's question on two largest orbits and the dihedral group of order eight (Kourovka Problem 18.44)

1 fully proved1 ready to prove

The dependency explorer needs JavaScript and its graph data.

Download the complete graph data.