Group-ring powers in fields

DefinitionGroup-ring powers and invariant elements (Kourovka 11.79)#✓fully proved

The proposition that the following holds for every prime \(p\), every infinite field \(F\) of characteristic \(p\), every finite group \(G\) of ring automorphisms of \(F\), and every element \(\xi = \sum_{g \in G} \xi_g \, g\) of the group ring \(\mathbb{Z}G\) with \(\xi \notin p\mathbb{Z}G\), that is, with some integer coefficient \(\xi_g\) not divisible by \(p\). For nonzero \(x \in F\) put \(x^\xi = \prod_{g \in G} g(x)^{\xi_g}\). If a subfield \(K\) of \(F\) contains \(x^\xi\) for every nonzero \(x \in F\), then \(K\) contains infinitely many elements \(y\) with \(g(y) = y\) for all \(g \in G\).

def subfieldContainsInfinitelyManyInvariants : Prop :=
  ∀ (p : ℕ) (F : Type) [Field F] [CharP F p], p.Prime → Infinite F →
    ∀ G : Subgroup (RingAut F), (G : Set (RingAut F)).Finite →
    ∀ ξ : G →₀ ℤ, (¬ ∀ g : G, (p : ℤ) ∣ ξ g) →
    ∀ K : Subfield F, (∀ x : F, x ≠ 0 → (ξ.prod fun g n => ((g : RingAut F) x) ^ n) ∈ K) →
      {y : F | y ∈ K ∧ ∀ g ∈ G, g y = y}.Infinite

Used by 1

Used bySubfields containing all group-ring powers have infinitely many invariants

TheoremSubfields containing all group-ring powers have infinitely many invariants#●ready to prove

Let \(p\) be a prime, \(F\) an infinite field of characteristic \(p\), \(G\) a finite group of ring automorphisms of \(F\), and \(\xi = \sum_{g \in G} \xi_g \, g\) an element of the group ring \(\mathbb{Z}G\) with \(\xi \notin p\mathbb{Z}G\), that is, with some integer coefficient \(\xi_g\) not divisible by \(p\). For nonzero \(x \in F\) put \(x^\xi = \prod_{g \in G} g(x)^{\xi_g}\). If a subfield \(K\) of \(F\) contains \(x^\xi\) for every nonzero \(x \in F\), then \(K\) contains infinitely many elements \(y\) with \(g(y) = y\) for all \(g \in G\). This is the positive answer to Kourovka Notebook Problem 11.79, given by K. N. Ponomarëv (1992).

public theorem_wanted subfieldContainsInfinitelyManyInvariants_holds :
    subfieldContainsInfinitelyManyInvariants

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 11.79 (K. N. Ponomarëv); resolution: K. N. Ponomarëv, Siberian Math. J. 33 (1992), 1094-1099, Kourovka Archive, Problem 11.79: "Yes", https://arxiv.org/abs/1401.0300v49.

Uses 1

Statement usesGroup-ring powers and invariant elements (Kourovka 11.79)

1 fully proved1 ready to prove

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