The L_2(11)-subalgebra of the Monster algebra

Sophie Decelle

Ars Mathematica Contemporanea, Tome 7 (2014) no. 1, pp. 83-103.

Résumé

We study a subalgebra V of the Monster algebra, VM, generated by three Majorana axes ax,  ay and az indexed by the 2A-involutions x,  y and z of M, the Monster simple group. We use the notation V =   ax, ay, az >> . 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 = CM(G) is isomorphic to the Mathieu group M12. For any pair {t,  s} of 2A-involutions, the pair of Majorana axes {at,  as} generates the dihedral subalgebra  at, as >> of VM, whose structure has been described in . In particular, the subalgebra  at, as >> contains the Majorana axis atst by the conjugacy property of dihedral subalgebras. Hence from the structure of its dihedral subalgebras, V coincides with the subalgebra of VM generated by the set of Majorana axes {at ∣ 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 { at ⋅ as ∣ s,  t ∈ T }, and with CVM(K) = V ⊕ ιM, where ιM is the identity of VM. 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).

Détail

DOI : 10.26493/1855-3974.259.0b4

Keywords: Monster algebra, Majorana, L2(11)