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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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Nonsingular systems of equations over arbitrary groups
Oct 5, 2026·nonsingular-systems-over-arbitrary-groups.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.
The Kervaire theorem for groups
Sep 24, 2026·The-Kervaire-Theorem-for-Groups-September-24-2026.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.