Download PDF by Buekenhout F., Huybrechts C.: A (c. L* )-Geometry for the Sporadic Group J2

By Buekenhout F., Huybrechts C.

We turn out the lifestyles of a rank 3 geometry admitting the Hall-Janko staff J2 as flag-transitive automorphism team and Aut(J2) as complete automorphism workforce. This geometry belongs to the diagram (c·L*) and its nontrivial residues are entire graphs of dimension 10 and twin Hermitian unitals of order three.

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The Basic Examples the multiplicity of 2, is 7-not 4, as one might expect. Compare also Berard Bergery-Bourguignon [I]. We now consider the real projective spaces. The n-dimensional real projective space P" is obtained by identifying antipodal points of 9,that is, S" covers P" with deck transformation group Z, = {IWn+ I S",-IW,+I S"}, where I,,,, is the identity map of [wnf1. Thus - &(Pn) = I Z k ( 9 ) = 2k(2k + n - I), with eigenspace consisting of homogeneous harmonic polynomials on R"+l, of degree 2k, restricted to s".

But here it is easy to show that the orthogonal group O(n, 1) preserving the quadratic form (19), not only leaves the hypersurface invariant-and is therefore an isometry of the hypersurface onto itself, but also acts transitively on the hypersurface, that is, given any two points pl, p z in the hypersurface, there exists an element in O(n, 1) which determines an isometry of the hypersurface and maps p1 onto p 2 . Using the transitivity of the action of O(n, l), one obtains the existence of a coordinate system for which (18) is valid about any point in the hyperbolic space.

All the basic facts in Section 3 about solutions to the eigenvalue problems remain unchanged except, of course, the differentiability of the solutions at the singularity of the boundary. We also note, in this case, that the functions, which are C" on M and the smooth part of aM, are dense in X ( M ) . To distinguish between the cases when aM is C", and when aM is possibly only piecewise C", we refer to connected M with compact closure and nonempty 16 1. The Laplacian C“ boundary as a regular domain, and to connected M with compact closure and nonempty piecewise C“ boundary as a normal domain.

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A (c. L* )-Geometry for the Sporadic Group J2 by Buekenhout F., Huybrechts C.

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