Suppose that a group G has socle L a simple large-rank classical group. Suppose furthermore that G acts transitively on the set of lines of a linear space S. We prove that, provided L has dimension at least 25, then G acts transitively on the set of flags of S and hence the action is known. For particular families of classical groups our results hold for dimension smaller than 25. The group the...

In this note we establish a (non-trivial) lower bound on the degree two entry h2 of a Gorenstein h-vector of any given socle degree e and any codimension r. In particular, when e = 4, that is for Gorenstein h-vectors of the form h = (1, r, h2, r, 1), our lower bound allows us to prove a conjecture of Stanley on the order of magnitude of the minimum value, say f(r), that h2 may assume. In fact, ...

Let 0 be a simple graph and let G be a group of automorphisms of 0. The graph is (G, 2)-arc transitive if G is transitive on the set of the 2-arcs of 0. In this paper we construct a new family of (PSU(3, q2), 2)-arc transitive graphs 0 of valency 9 such that Aut0 = Z3.G, for some almost simple group G with socle PSU(3, q2). This gives a new infinite family of non-quasiprimitive almost simple gr...

We characterize Frobenius algebras A as algebras having a comultiplication which is a map of A-modules. This characterization allows a simple demonstration of the compatibility of Frobenius algebra structure with direct sums. We then classify the indecomposable Frobenius algebras as being either \annihilator algebras" | algebras whose socle is a principal ideal | or eld extensions. The relation...

Let G be a finite group and cd(G) denote the set of complex irreducible character degrees of G. In this paper, we prove that if G is a finite group and H is an almost simple group whose socle is a Mathieu group such that cd(G) = cd(H), then there exists an abelian subgroup A of G such that G/A is isomorphic to H. In view of Huppert’s conjecture (2000), we also provide some examples to show that...

We study a number of conditions on the Hilbert function of a level artinian algebra which imply the Weak Lefschetz Property (WLP). Possibly the most important open case is whether a codimension 3 SI-sequence forces the WLP for level algebras. In other words, does every codimension 3 Gorenstein algebra have the WLP? We give some new partial answers to this old question: we prove an affirmative a...