kaplansky_idempotent_conjecture

⚠️ Retired — specification defect

The Lean statement below does not encode the problem shown on this page, so its Proved status carries no information about that problem. Do not import this node or use it as a dependency.

Kaplansky's idempotent conjecture (1956): The group algebra ℂ[G] of a torsion-free group G has no nontrivial idempotents (elements e ≠ 0, 1 with e²=e). Proved for many classes (amenable groups, virtually solvable groups). General case open; related to the zero divisor conjecture.

Why this node was retired

The posted statement is

import Mathlib

theorem kaplansky_idempotent_conjecture (G : Type*) [Group G]
    (htf : ∀ g : G, ∀ n : ℕ, g ^ n = 1 → g = 1) :
    ∀ (a : MonoidAlgebra ℂ G),
      a ^ 2 = a →
      a = 0 ∨ a = 1 := by
  sorry

The purported torsion-freeness hypothesis quantifies over n=0, forcing every element of G to be 1.

A proof of a malformed proposition can be a correct proof of that proposition, so this is not a judgment on the accepted submission — but the Proved status must not be read as settling the problem shown above.

Proposed corrected statement

Use torsion-freeness ∀g∈G,∀n∈ℕ,0<n → g^n=1 → g=1. For the idempotent target conjecture a²=a → a=0∨a=1 in ℂ[G]; for the zero-divisor target conjecture ab=0 → a=0∨b=0.

Diagnosis and correction from the public Prove2Me statement audit (wamlat/prove2me-errors). The correction is natural-language mathematics and is not Lean-verified — it is a specification for a corrected node, not a drop-in replacement. No corrected replacement node exists yet.