Fitting's Theorem and Semirings of Normal Subgroups

2026-07-31

AI Summary

This work reinterprets Fitting’s theorem—which states that the product of two nilpotent normal subgroups is again nilpotent—from the perspective of algebraic structure. To this end, it introduces a commutative, generally non-associative semiring without identity on the set of all normal subgroups of a group \( G \), thereby embedding Fitting’s theorem for the first time within a semiring framework. The argument relies crucially on a binomial expansion formula adapted to non-associative settings and inclusion relations involving commutators of normal subgroups. The entire theoretical development has been formally verified in the Lean proof assistant using the Mathlib library, achieving a rigorous reconstruction and machine-checked validation of this classical result in a novel algebraic context.

Abstract

We define a non-unital, generally non-associative, commutative semiring structure on the collection of normal subgroups of a group $G$. This viewpoint allows us to recast in ring-theoretic terms Fitting's classical theorem that the join of two nilpotent normal subgroups is nilpotent. From this perspective, the two key inputs are a binomial expansion in a non-associative setting and the fact that the commutator subgroup of two normal subgroups lies in each factor. The development is formalized in Lean, making essential use of Mathlib for the core definitions and results.

Problem

• Fitting's Theorem
• normal subgroups
• semiring
• nilpotent groups
• commutator subgroup

Innovation

• semiring
• normal subgroups
• Fitting's theorem
• non-associative algebra
• formalization in Lean