On the Structure of Overgroup Lattices in Exceptional Groups of Type G2(q)
نویسنده
چکیده
We study overgroup lattices in groups of type G2 over finite fields and reduce Shareshians conjecture on D∆-lattices in the G2 case to a smaller question about the number of maximal elements that can be conjugate in such a lattice. Here we establish that, modulo this smaller question, no sufficiently large D∆-lattice is isomorphic to the lattice of overgroups OG(H) in G = G2(q) for any prime power q and any H ≤ G. This has application to the question of Palfy and Pudlak as to whether each finite lattice is isomorphic to some overgroup lattice inside some finite group.
منابع مشابه
On Conjugacy Separability of Fibre Products
In this paper we study conjugacy separability of subdirect products of two free (or hyperbolic) groups. We establish necessary and sufficient criteria and apply them to fibre products to produce a finitely presented group G1 in which all finite index subgroups are conjugacy separable, but which has an index 2 overgroup that is not conjugacy separable. Conversely, we construct a finitely present...
متن کاملEigenvalues of the Cayley Graph of Some Groups with respect to a Normal Subset
Set X = { M11, M12, M22, M23, M24, Zn, T4n, SD8n, Sz(q), G2(q), V8n}, where M11, M12, M22, M23, M24 are Mathieu groups and Zn, T4n, SD8n, Sz(q), G2(q) and V8n denote the cyclic, dicyclic, semi-dihedral, Suzuki, Ree and a group of order 8n presented by V8n = < a, b | a^{2n} = b^{4} = e, aba = b^{-1}, ab^{...
متن کاملQUANTALE-VALUED SUP-ALGEBRAS
Based on the notion of $Q$-sup-lattices (a fuzzy counterpart of complete join-semilattices valuated in a commutative quantale), we present the concept of $Q$-sup-algebras -- $Q$-sup-lattices endowed with a collection of finitary operations compatible with the fuzzy joins. Similarly to the crisp case investigated in cite{zhang-laan}, we characterize their subalgebras and quotients, and following...
متن کاملStructure and representations of exceptional groups
From Cartan and Killing’s original classification of simple Lie groups in the 1890s, these groups have been understood to be of two rather different types: the infinite families of classical groups (related to classical linear algebra and geometry); and a finite number of exceptional groups, ranging from the 14-dimensional groups of typeG2 to the 248-dimensional groups of typeE8. Often it is po...
متن کاملPreventive Effect of Novel Bacterial Polysaccharide and Animal Splenic Protein as Natural Adjuvants on Animal Model of Asthma
Objective(s) Two new adjuvants from natural animal lipids (G2) and bacterial polysaccharide extracts (PC) were previously prepared by our group and showed a reduction in tracheal responsiveness. The aim of this study was to evaluate the preventive effect of recently introduced natural products (G2 and PC) on the development of asthma. Materials and Methods Asthma was induced using a standard...
متن کامل