Totient sums of element orders

DefinitionCurtin–Pourgholi question on totient sums and simplicity (Kourovka 19.25)#✓fully proved

The statement that for all finite groups \(G\) and \(H\) of the same order with \(\sum_{g \in G} \varphi(|g|) = \sum_{h \in H} \varphi(|h|)\), where \(|g|\) is the order of \(g\) and \(\varphi\) is Euler's totient function, if \(G\) is simple then \(H\) is simple.

def sameOrderTotientSumSimple : Prop :=
  ∀ (G H : Type) [Group G] [Group H] [Fintype G] [Fintype H],
    Fintype.card G = Fintype.card H →
    ∑ g : G, Nat.totient (orderOf g) = ∑ h : H, Nat.totient (orderOf h) →
    IsSimpleGroup G → IsSimpleGroup H

Used by 1

Used byOrder and totient sum do not determine simplicity (Kourovka 19.25)

TheoremOrder and totient sum do not determine simplicity (Kourovka 19.25)#●ready to prove

There are finite groups \(G\) and \(H\) of the same order with \(\sum_{g \in G} \varphi(|g|) = \sum_{h \in H} \varphi(|h|)\), where \(|g|\) is the order of \(g\) and \(\varphi\) is Euler's totient function, such that \(G\) is simple and \(H\) is not simple. So the Curtin–Pourgholi question has a negative answer.

Van Doorn, Judin, Monticone and Morrison (Theorem 4.1) take for \(G\) the projective special unitary group \(\mathrm{PSU}(3, 3)\), which is simple of order \(6048\), and \(H = C_6 \times S_4 \times F\), where \(C_6\) is the cyclic group of order \(6\), \(S_4\) is the symmetric group on four letters and \(F\) is the Frobenius group of order \(42\); both totient sums equal \(23984\).

public theorem_wanted sameOrderTotientSumSimple_refuted : ¬ sameOrderTotientSumSimple

Sources

• E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Problem 19.25 (B. Curtin, G. R. Pourgholi); resolution: P. Monticone, Kourovka Notebook Problem 19.25, solution note (2026); W. van Doorn, E. Judin, P. Monticone and D. Morrison, On Some Problems from the Kourovka Notebook, preprint, arXiv:2607.17477 (2026), Section 4, Theorem 4.1 with Lemma 4.2 (G = PSU(3, 3) is simple, by Iwasawa's criterion, of order 6048 with totient sum 23984) and Lemma 4.3 (H = C₆ × S₄ × F, F the Frobenius group of order 42, has the same order and totient sum and is not simple); the solution was found and verified in Lean by the AI system Aristotle (abstract and Appendix A); the Kourovka Notebook (21st issue) marks 19.25 solved: No, it need not be (AI solution confirmed by GAP), https://arxiv.org/abs/2607.17477.

Uses 1

Statement usesCurtin–Pourgholi question on totient sums and simplicity (Kourovka 19.25)

1 fully proved1 ready to prove

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