The L_2(11)-subalgebra of the Monster algebra

Abstract

We study a subalgebra V of the Monster algebra, V M , generated by three Majorana axes a x , a y and a z indexed by the 2 A -involutions x , y and z of M , the Monster simple group. We use the notation V = << a x , a y , a z >> . We assume that x y is another 2 A -involution and that each of x z , y z and x y z has order 5 . Thus a subgroup G of M generated by { x , y , z } is a non-trivial quotient of the group G (5, 5, 5) = < x , y , z ∣ x 2 , y 2 , ( x y ) 2 , z 2 , ( x z ) 5 , ( y z ) 5 , ( x y z ) 5 > . It is known that G (5, 5, 5) is isomorphic to the projective special linear group L 2 (11) which is simple, so that G is isomorphic to L 2 (11) . It was proved by S. Norton that (up to conjugacy) G is the unique 2 A -generated L 2 (11) -subgroup of M and that K = C M ( G ) is isomorphic to the Mathieu group M 12 . For any pair { t , s } of 2 A -involutions, the pair of Majorana axes { a t , a s } generates the dihedral subalgebra << a t , a s >> of V M , whose structure has been described in . In particular, the subalgebra << a t , a s >> contains the Majorana axis a t s t by the conjugacy property of dihedral subalgebras. Hence from the structure of its dihedral subalgebras, V coincides with the subalgebra of V M generated by the set of Majorana axes { a t ∣ t ∈ T } , indexed by the 55 elements of the unique conjugacy class T of involutions of G ≅ L 2 (11) . We prove that V is 101 -dimensional, linearly spanned by the set { a t ⋅ a s ∣ s , t ∈ T } , and with C V M ( K ) = V ⊕ ι M , where ι M is the identity of V M . Lastly we present a recent result of Á. Seress proving that V is equal to the algebra of the unique Majorana representation of L 2 (11) .