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 2A-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 xy is another 2A-involution and that each of xz,  yz and xyz 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 ∣ x²,  y²,  (xy)²,  z²,  (xz)⁵,  (yz)⁵, (xyz)⁵ > . It is known that G^{(5, 5, 5)} is isomorphic to the projective special linear group L₂(11) which is simple, so that G is isomorphic to L₂(11). It was proved by S. Norton that (up to conjugacy) G is the unique 2A-generated L₂(11)-subgroup of M and that K = C_M(G) is isomorphic to the Mathieu group M₁₂. For any pair {t,  s} of 2A-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_tst 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 ≅ L₂(11). We prove that V is 101-dimensional, linearly spanned by the set { a_t ⋅ a_s ∣ s,  t ∈ T }, and with C_VM(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 L₂(11).