Lie algebras and 3-transpositions

TL;DR

This paper presents a construction linking 3-transpositions in groups to Lie algebras over GF(2), revealing connections to sporadic groups and classical Lie algebras.

Contribution

It introduces a novel algebraic construction from 3-transpositions that characterizes certain simple Lie algebras and embeds sporadic groups into classical groups.

Findings

Identifies which simple Lie algebras can be constructed from 3-transpositions.

Provides a natural embedding of the sporadic group Fi22 into ^2E_6(2).

Establishes a new link between group theory and Lie algebra theory.

Abstract

We describe a construction of an algebra over the field of order 2 starting from a conjugacy class of 3-transpositions in a group. In particular, we determine which simple Lie algebras arise by this construction. Among other things, this construction yields a natural embedding of the sporadic simple group $\Fi{22}$ in the group $^2E_6(2)$.