This article is a contribution to the study of the automorphism groups of 2 − (v, k, 1) designs. Let G act as a block-transitive and point-primitive automorphism group of a non-trivial design D with v points and blocks of size k. Set k2 = (k, v−1). Assume q = pf for some prime p and positive integer f . If q ≥ [(k2k − k2 + 1)f ]2, then Soc(G), the socle of G, is not PSL(2, q). Mathematics Subje...

In this paper we study standard graded artinian level algebras, in particular those whose socle-vector has type 2. Our main results are: the characterization of the level h-vectors of the form (1, r, ..., r, 2) for r ≤ 4; the characterization of the minimal free resolutions associated to each of the h-vectors above when r = 3; a sharp upper-bound (under certain mild hypotheses) for the level h-...

We give a description of infinite families of finite primitive permutation groups for which there is a uniform finite upper bound on the diameter of all orbital graphs. This is equivalent to describing families of finite permutation groups such that every ultraproduct of the family is primitive. A key result is that, in the almost simple case with socle of fixed Lie rank, apart from very specif...

Let G be a transitive permutation group on a finite set Ω of size at least 2. An element of G is a derangement if it has no fixed points on Ω. Let r be a prime divisor of |Ω|. We say that G is r-elusive if it does not contain a derangement of order r, and strongly r-elusive if it does not contain one of r-power order. In this note we determine the r-elusive and strongly r-elusive primitive acti...

We show that bigrassmannian permutations determine the socle of cokernel an inclusion Verma modules in type $A$. All such socular constituents turn out to be indexed by Weyl group elements from penultimate two-sided cell. Combinatorially, a module $w\in S_n$ into dominant are shown determined essential set $w$ and their degrees graded picture computable terms associated rank function. As applic...

An interesting fact is that almost all the connected 2-arc-transitive nonnormal Cayley graphs on nonabelian simple groups with small valency or prime (provided solvable vertex stabilizers) are alternating An. This naturally motivates study of for arbitrary valency. In this paper, we characterize automorphism such graphs. particular, show a non-complete (G,2)-arc-transitive graph G simple, socle...

Let A be a type t artinian level algebra with h-vector h =) be the h-vector of the generic type c level quotient of A having the same socle degree e. The main result of this paper is an (in general sharp) lower-bound for the h-vector H H c,gen u ≥ 1 t 2 − 1 ((t − c)h e−u + (ct − 1)h u). This result generalizes a recent theorem of Iarrobino (which treats the case t = 2). Finally, we begin to obt...

The socle of a graded Buchsbaum module is studied and is related to its local cohomology modules. This algebraic result is then applied to face enumeration of Buchsbaum simplicial complexes and posets. In particular, new necessary conditions on face numbers and Betti numbers of such complexes and posets are established. These conditions are used to settle in the affirmative Kühnel’s conjecture ...

A new characteristic property of von Neumann regular rings is proposed in terms of annihilators of elements. An ELT fully idempotent ring is a regular ring whose simple left (or right) modules are either injective or projective. Artinian rings are characterized in terms of Noetherian rings. Strongly regular rings and rings whose two-sided ideals are generated by central idempotents are characte...

Almost all primitive permutation groups of degree n have order at most n · Q[log2 n]−1 i=0 (n − 2) < n2 , or have socle isomorphic to a direct power of some alternating group. The Mathieu groups, M11, M12, M23 and M24 are the four exceptions. As a corollary the sharp version of a theorem of Praeger and Saxl is established, where M12 turns out to be the ”largest” primitive group. For an applicat...