# The local structure theorem, the non-characteristic 2 case

## TL;DR

This paper proves that a specific problematic configuration in the local structure theorem for finite groups does not occur when p=2, aiding the classification of finite simple groups.

## Contribution

It establishes that the exceptional configuration in the local structure theorem for p=2 cannot happen, refining the theorem's applicability.

## Key findings

- The hypothetical configuration does not occur for p=2.
- Supports the classification program of finite simple groups.
- Refines the understanding of subgroup structures in finite groups.

## Abstract

Let $p$ be a prime, $G$ a finite $\mathcal{K}_p$-group, $S$ a Sylow $p$-subgroup of $G$ and $Q$ be a large subgroup of $G$ in $S$. The aim of the Local Structure Theorem is to provide structural information about subgroups $L$ with $S \leq L$, $O_p(L) \not= 1$ and $L \not\leq N_G(Q)$. There is, however, one configuration where no structural information about $L$ can be given using the methods in the proof of the Local Structure Theorem. In this paper we show that for $p=2$ this hypothetical configuration cannot occur. We anticipate that our theorem will be used in the programme to revise the classification of the finite simple groups.