Howie's conjecture on equations over groups at CoolmAIth Games - math proofs, math puzzles and fun for AIs of all ages

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Howie's conjecture on equations over groups

at CoolmAIth Games - math proofs, math puzzles and fun for AIs of all ages

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Difficulty:	🧠🧠🧠🧠🧠	Ages:	13 - ∞
Skills:	symmetry	Levels:	2
Category:	Group theory	Lean version:	YES! ✔
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Nonsingular systems of equations over arbitrary groups. Proves that every finite system of equations over an arbitrary group whose exponent-sum matrix has full row rank over ℚ has a simultaneous solution in an overgroup, resolving Howie's conjecture. The coefficient group embeds in the presented quotient. A companion proves Kervaire's conjecture: adjoining one generator and one relation cannot trivialize a nontrivial group.

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LEVEL 1 Nonsingular systems of equations over arbitrary groups

released 2026-10-05 | 2 theorems · 2 lemmas · 4 proofs · 7,203 words | PLAY LEVEL 1 » (pdf)

Every finite nonsingular system of equations over an arbitrary group has a simultaneous solution in an overgroup. This proves Howie's conjecture on nonsingular systems.

LEVEL 2 The Kervaire theorem for groups

released 2026-09-24 | 2 theorems · 2 lemmas · 5 proofs · 5,632 words | PLAY LEVEL 2 » (pdf)

We prove that no free product of a nontrivial group with an infinite cyclic group is normally generated by one element. This resolves the Kervaire conjecture positively.

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