Monster embeddings of certain 3-transposition groups via Majorana representations

In this paper, we study the Majorana representations of symplectic and orthogonal groups over <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>GF</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mn>2</m:mn> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> GF(2) . We show that such a group admits a Majorana representation only if it can be embedded into the Monster group as a <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>2</m:mn> <m:mo>⁢</m:mo> <m:mi>A</m:mi> </m:mrow> </m:math> 2A -generated subgroup.