We prove that a kG-module has a semilinear tensor decomposition if and only if its endomorphism algebra has a pair of mutually centralizing, unital, G-invariant subalgebras that are not commutative and are isomorphic to complete matrix algebras over an extension field K of k. We give an algorithm that constructs a semilinear tensor decomposition for any module whose endomorphism algebra contains appropriate invariant subalgebras.

Condensation is a technique that can often predict a Brauer character table of a finite group with a very high degree of confidence, but without a proof of correctness. In this paper we describe a strategy that can give such a proof. We introduce and apply two novel condensation methods: virtual tensor condensation and the condensation of bilinear forms. We illustrate our strategy and new techniques with examples taken from our computation of the 5-modular Brauer character table of the sporadic simple Lyons group.

John Conway challenged and inspired us in many different ways. His influence can be seen across so many areas of mathematics that a complete account would fill many volumes, which would continue to expand forever. These three articles present very different parts of mathematics where modern approaches are built on foundations laid by John.

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 one of these properties. Our algorithm reduces the computation to an instance of the Pure Tensor problem. This asks whether a subspace of a tensor product of vector spaces X and Y contains an element of the form x⊗y with x∈X and y∈Y. We show that the Pure Tensor Problem reduces to the calculation of an appropriate Gröbner basis.

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.

John Horton Conway lived to discover the Mathematics behind problems, always working to isolate a pure, essential kernel of truth. He loved to communicate these simple truths to others, often changing the way they thought. Everything John touched turned to Mathematics, and to very beautiful Mathematics. John was generous with his mathematical riches; he gave them to everyone that showed interest - whether at the coffee house, at the sushi restaurant, or in Mathematics departments. He was a magnet for all mathematicians, and he welcomed all who came to him. John would find a way to start with some simple calculation, a game or puzzle and turn it into whatever he wanted to explain. John often chose problems about games and recreational topics, but the insights he derived changed our understanding of several very serious branches of Mathematics.

We compute the 3-modular character table of the group 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 O'N . We finish the remaining questions module theoretically with an application of condensation.

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 avoid such problems.

We discuss conjugacy classes of embeddings of Alternating groups in Exceptional Lie groups. We settle the count of classes of embeddings in E-8 of a subgroup Alt(10) and its double cover. This involves computation and the reduction of the problems to relative eigenvector problems. We update previously published tables of embeddings. We comment on the improvements present in our table and on the remaining unsettled conjugacy questions.

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 ouvrages de mathematique et de physique par Messieurs de l'Academie Royale des Sciences. In his first paper, “Des Quarrez ou Tables Magiques,” Frenicle quotes a rule for constructing magic squares of odd order. However, Frenicle is more famous for his second paper, “Table Generale des Quarrez de Quatres,” in which he enumerates the 880 magic squares of order four. His enumeration has been repeated many times. These later enumerations they have confirmed the remarkable fact that he was correct.

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 sphere centred at O. The typical side is a = BC is a great circle arc from to that lies in the plane OBC; its length is the angle subtended at O. Similarly, the typical angle between the two sides AB and AC is the angle between the planes OAB and OAC.

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 mathematicians, in particular to the great Swiss geometer Jakob Steiner, who provided a proof. In this paper we give several proofs and discuss the old query: ‘Is there a direct proof?’ before suggesting that this is no longer the right question to ask. We go on to discuss all cases when an angle bisector (internal orexternal) of some angle is equal to one of another.

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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 boards the domination number gives the size of a minimal X-pentomino covering. Natural sequences that count maximal saturated domino coverings of square and rectangular boards are obtained. These include the new sequences A193764, A193765, A193766, A193767, and A193768 of OEIS.

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 incenter and excenters subject to fixed locations for the circumcenter and orthocenter.

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