FormalConjectures.Kourovka.«20

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import FormalConjecturesUtil

Conjecture 20.76

by L. Pyber Reference: The Kourovka Notebook !

namespace Kourovka.«20.76»

Let $G$ be a finite $p$-group and assume that all abelian normal subgroups of $G$ have order at most $p^k$. Is it true that every abelian subgroup of $G$ has order at most $p^{2k}$?

@[category research open, AMS 20]
theorem declaration uses 'sorry'kourovka.«20.76» : answer(sorry) ↔
    ∀ᵉ (p : ℕ) (hp : p.Prime) (G : Type) (_ : Group G) (hg : IsPGroup p G) (_ : Finite G) (k : ℕ)
    (h : ∀ H: Subgroup G, H.Normal ∧ IsMulCommutative H → Nat.card H ≤ p ^ k),
    (∀ H : Subgroup G, IsMulCommutative H → Nat.card H ≤ p ^ (2 * k)) := by⊢ True ↔
  ∀ (p : ℕ),
    Nat.Prime p →
      ∀ (G : Type) (x : Group G),
        IsPGroup p G →
          Finite G →
            ∀ (k : ℕ),
              (∀ (H : Subgroup G), H.Normal ∧ IsMulCommutative ↥H → Nat.card ↥H ≤ p ^ k) →
                ∀ (H : Subgroup G), IsMulCommutative ↥H → Nat.card ↥H ≤ p ^ (2 * k)
  sorryAll goals completed! 🐙


end Kourovka.«20.76»