Alex Ryba

• Queens College, CUNY

Let V be an irreducible module for a finite group G over an algebraically closed field k‾. We prove that the algebra Homk‾(V,V) has a proper G-invariant subalgebra if and only if V is either imprimitive or tensor decomposable. We give an algorithm to determine whether an absolutely irreducible matrix representation defined over a finite field has o...

The Pascal Multimysticum is a system of points and lines constructed with a straight edge starting from six points on a conic. We show that the system contains 150 infinite ranges (and 150 infinite pencils) whose projective coordinates are absolutely fixed and independent of the conic and the hexagon that define the system.

We compute the 13-modular character table of the bicyclic extension 2.Suz.2 given in the Atlas. This completes the last unknown modular character table of a bicyclic extension of the sporadic simple group Suz. We explain an issue of compatibility between different modular tables and partially recompute the 7-modular table of 2.Suz.2 in order to avo...

We compute the 3-modular character table of the group \(\mathrm{O'N}.2\). Much of the table is deduced character theoretically from the known 3-modular character table of the sporadic simple O’Nan group \(\text {O}'\text {N}\). We finish the remaining questions module theoretically with an application of condensation.

This chapter discusses magic squares. A magic square of order n is an arrangement of the numbers from 1 to n ² in an n × n array so that the two diagonals and all the rows and columns have the same sum. This sum is called the magic constant. Bernard Frenicle de Bessy's work on magic squares appears in two papers published in the book Divers ouvrage...

Although high school textbooks from early in the 20th century show that spherical trigonometry was still widely taught then, today very few mathematicians have any familiarity with the subject. The first thing to understand is that all six parts of a spherical triangle are really angles — see Figure 1. This shows a spherical triangle ABC on a sphe...

In 1840 C. L. Lehmus sent the following problem to Charles Sturm: ‘If two angle bisectors of a triangle have equal length, is the triangle necessarily isosceles?’ The answer is ‘yes’, and indeed we have the reverse-comparison theorem : Of two unequal angles, the larger has the shorter bisector (see [1, 2]). Sturm passed the problem on to other math...

A domino covering of a board is saturated if no domino is redundant. We introduce the concept of a fragment tiling and show that a minimal fragment tiling always corresponds to a maximal saturated domino covering. The size of a minimal fragment tiling is the domination number of the board. We define a class of regular boards and show that for these...

We strengthen a result of Lehmer, obtaining a new necessary condition for the roots of a complex polynomial to have equal modulus. From this we derive the famous theorem of Feuerbach, as well as the less well-known theorems of Euler and Guinand on the tritangent centers of a triangle. The latter theorems constrain the possible locations of the ince...

We present an algorithm to reduce the constructive membership problem for a black-box group G to three instances of the same problem for involution centralisers in G. If G is a simple group of Lie type in odd characteristic, then this reduction can be per formed in (Monte Carlo) polynomial time.

We determine the 5-modular character table of the sporadic simple Harada–Norton group HNHN and its automorphism group HN.2HN.2 using The Meat-Axe and condensation.

We give a polynomial time algorithm that finds a split Cartan subalgebra of a finite Chevalley Lie algebra.

We give a polynomial time algorithm that replaces an arbitrary module of a finite Chevalley group by the adjoint module written with respect to a Chevalley basis.

We construct two embeddings of finite groups into groups of Lie type. These embeddings have the interesting property that the finite subgroup acts irreducibly on a minimal module for the group of Lie type. We present our constructions as examples of a general method that obtains embeddings into groups of Lie type.

An algorithm for condensing symmetrized tensor powers of a modular representation of a finite group with respect to a monomial subgroup is introduced. We show that our algorithm provides a practical tool for analyzing tensor powers of modules. We include a brief review of symmetrization of tensor powers and of condensation of modules.

We classify embeddings of the finite simple groups PSL(2, 41) andPSL (2, 49) into algebraic groups of type E8in characteristic not dividing the orders of these respective groups. Lie theory and finite group theory point to families of elements in a group of typeE8 over a finite field which might satisfy a presentation for our finite simple group. T...

Since the early 1980s, there have been efforts to determine which finite simple groups have a projective embedding into an exceptional complex algebraic group, i.e., one of G2(ℂ), F4(C), 3E6(ℂ), 2E7(ℂ), E8(ℂ) (note that these are simply connected groups). In this article, we settle exactly which finite quasisimple groups embed in each exceptional a...

We give a non-enumerative proof that the P -positions of the hexad game give the Steiner system S(5; 6; 12). We show that the distributions of nim sums and Welter values of these P -positions have simple and surprising regularities.

Since the mid 80s, there have been efforts to classify projective embeddings of finite simple groups in the exceptional complex algebraic groups, G2(ℂ), F4(ℂ), E6(ℂ), E7(ℂ), E8(ℂ). In this article, we settle the final unresolved question about which finite simple groups have at least one such projective embedding. This result was announced in [5]....

In this paper we describe our discovery that the sporadic simple groups HS and M22 are contained in the simple Chevalley group E7(5). The work of [9] produces a short list of the possibilities for a sporadic simple subgroup of an exceptional group of Lie type. Apart from possible embeddings of M22 and HS in groups of type E7 in characteristic 5, a...

Let L be the A1 root lattice and let G be a finite subgroup of Au(V), where V = VL is the associated lattice VOA (in this case, Aut(V) ≅ PSL(2, )). The fixed point sub-VOA, VG, was studied previously by the authors, who found a set of generators and determined the automorphism group when G is cyclic (from the “A-series”) or dihedral (from the “D-se...

Since nite simple groups are the building blocks of nite groups, it is natural to ask about their occurrence \in nature". In this article, we consider their occurrence in algebraic groups and moreover discuss the general theory of nite subgroups of algebraic groups. summary. The classication program for nite subgroups of complex algebraic groups in...

The Harada-Norton group is one of the twenty-six sporadic simple groups. It has order 273, 030, 912, 000, 000 = 214.36.56.7.11.19. In this paper our main objective is: Theorem 1. The Harada-Norton group acts as a group of linear automorphisms of a 133-dimensional commutative, non-associative algebra defined over F5.

Peer Reviewed http://deepblue.lib.umich.edu/bitstream/2027.42/46579/1/222_2005_Article_BF01231561.pdf

Condensation replaces an explicit matrix representation of a group by a related but much smaller representation of a Hecke algebra. We give a general account of condensation and show how condensation is used to obtain the structure of large matrix representations. We include a detailed description of a new implementation of condensation. As an exam...

We determine which known sporadic geometries have projective Lefschetz modules. An elementary lemma on local collapsibility greatly simplifies the task of verifying projectivity. The modules are analyzed when possible in terms of projective covers of individual irreducibles.