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OP-9744Status unassessedAlgebra

Conjecture 19.25

Conjecture 19.25

Specialist source · status unassessedAs of 2026-09-10Source & evidence ↓
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Problem statement

kourovka_19_25

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

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Formal statement

Lean 4
/-
Copyright 2026 The Formal Conjectures Authors.

Licensed under the Apache License, Version 2.0 (the "License");
you may not use this file except in compliance with the License.
You may obtain a copy of the License at

    https://www.apache.org/licenses/LICENSE-2.0

Unless required by applicable law or agreed to in writing, software
distributed under the License is distributed on an "AS IS" BASIS,
WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
See the License for the specific language governing permissions and
limitations under the License.
-/

import FormalConjecturesUtil

/-!
# Conjecture 19.25

by B. Curtin, G. R. Pourgholi

*Reference:* [The Kourovka Notebook](https://arxiv.org/abs/1401.0300v40)
-/

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 kourovka_19_25 : answer(sorry) ↔
    ∀ (G H : Type) [Group G] [Group H] [Fintype G] [Fintype H],
       Fintype.card G = Fintype.card H →
       ∑ g : G, φ (orderOf g) = ∑ h : H, φ (orderOf h) →
       IsSimpleGroup G → IsSimpleGroup H := by
  sorry

end Kourovka.«19.25»

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Background

Source snapshot inspected on 10 September 2026. Subsequent research status and the accuracy of source formalizations have not been independently certified.

Fixed commit b82b08faa9006484021c12005ab41287fb2ffb69. This file contains 1 source-labeled research-open declarations. Other declarations in the attached file can be definitions, solved results or tests. The file is preserved as source material; it has not been compiled or proved by this import.

Copyright The Formal Conjectures Authors. Code: Apache-2.0; other original material: CC BY 4.0. Third-party sources retain their own licenses. Original source files and notices preserved.

References
  1. [1]Formal Conjectures — Conjecture 19.25 link
  2. [2]Original reference cited in the formalization link
  3. [3]Kourovka — Problem 19.25 link

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