FormalConjectures.Kourovka.«19

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import FormalConjecturesUtil

Conjecture 19.25

by B. Curtin, G. R. Pourgholi

Reference: The Kourovka Notebook


open scoped Nat Group


namespace Kourovka.«19.25»

Let $G$ and $H$ be finite groups of the same order with $\sum_{g \in G} \phi(|g|) = \sum_{h \in H} \phi(|h|)$, where $\phi$ is the Euler totient function. Suppose that $G$ is simple. Is $H$ necessarily simple?

@[category research open, AMS 20]
theorem declaration uses 'sorry'kourovka.«19.25» : answer(sorry) ↔
    ∀ (G H : Type) [Group G] [Group H] [Fintype G] [Fintype H],
       ∑ g : G, φ (orderOf g) = ∑ h : H, φ (orderOf h) →
       IsSimpleGroup G → IsSimpleGroup H := by⊢ True ↔
  ∀ (G H : Type) [inst : Group G] [inst_1 : Group H] [inst_2 : Fintype G] [inst_3 : Fintype H],
    ∑ g, φ (orderOf g) = ∑ h, φ (orderOf h) → IsSimpleGroup G → IsSimpleGroup H
  sorryAll goals completed! 🐙


end Kourovka.«19.25»