Explicit Computation of a Galois Representation Attached to an Eigenform Over $${\text {SL}}_3$$ from the $${{\text {H}}}_{\acute{\mathrm{e}}\mathrm{t}}^2$$ of a Surface

We describe a method to compute mod $$\ell $$ ℓ Galois representations contained in the $${{\text {H}}}_{\acute{\mathrm{e}}\mathrm{t}}^2$$ H e ´ t 2 of surfaces. We apply this method to the case of a representation with values in $${\text {GL}}_3(\mathbb {F}_9)$$ GL 3 ( F 9 ) attached to an eigenform over a congruence subgroup of $${\text {SL}}_3$$ SL 3 . We obtain, in particular, a polynomial with Galois group isomorphic to the simple group $${\text {PSU}}_3(\mathbb {F}_9)$$ PSU 3 ( F 9 ) and ramified at 2 and 3 only.

Coleman Automorphisms of Extensions of Finite Characteristically Simple Groups by Some Finite Groups

Let [Formula: see text] be an extension of a finite characteristically simple group by an abelian group or a finite simple group. It is shown that every Coleman automorphism of [Formula: see text] is an inner automorphism. Interest in such automorphisms arises from the study of the normalizer problem for integral group rings.

Arc-Transitive Graphs of Square-free Order with Valency 11

In this paper, we present a complete list of connected arc-transitive graphs of square-free order with valency 11. The list includes the complete bipartite graph [Formula: see text], the normal Cayley graphs of dihedral groups and the graphs associated with the simple group [Formula: see text] and [Formula: see text], where [Formula: see text] is a prime.

Beyond the Standard Model

We review the fundamental physics questions left unanswered by the Standard Model, and we explain why, despite its great successes, the search for physics beyond the Standard Model is a very active field of research. We briefly review the theories of Grand Unification which assume that the SU(3) × SU(2) x U(1) group of the Standard Model is the remnant of a larger, simple or semi-simple group, spontaneously broken at very high energies. These theories predict the phenomenon of proton decay and we discuss possible cosmological consequences of such an instability. We end with the theory of supersymmetry which postulates the existence of an approximate fermion-boson symmetry.

COMPUTATIONAL INVESTIGATION OF A4-GRAPHS FOR CERTAIN LEECH LATTICE GROUPS

In the case of a finite simple group , and -conjugacy class of element of order 3, The A4-graph is define as simple graph denoted by A4 has as vertex set and are adjacent if and only if x≠y and xy-1 = yx-1.We aim to investigate computationally the structure of theA4 when Leech Lattice groups.

The Möbius function of PSL(3,2p) for any prime p

Let [Formula: see text] be the simple group [Formula: see text], where [Formula: see text] is a prime number. For any subgroup [Formula: see text] of [Formula: see text], we compute the Möbius function [Formula: see text] of [Formula: see text] in the subgroup lattice of [Formula: see text]. To this aim, we describe the intersections of maximal subgroups of [Formula: see text]. We point out some connections of the Möbius function with other combinatorial objects, and, in this context, we compute the reduced Euler characteristic of the order complex of the subposet of [Formula: see text]-subgroups of [Formula: see text], for any prime [Formula: see text] and any prime power [Formula: see text].