# The L_2(11)-subalgebra of the Monster algebra

## DOI:

https://doi.org/10.26493/1855-3974.259.0b4## Keywords:

Monster algebra, Majorana, L2(11)## 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*_{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* ≅ *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*_{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 *L*_{2}(11).

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