Determine the value of the variable in each equation.

Basic Algebra = 代数学入門

In this chapter we describe the six families of so-called ‘classical’ simple groups. These are the linear, unitary and symplectic groups, and the three families of orthogonal groups. All may be obtained from suitable matrix groups G by taking G′/Z(G′). Our main aims again are to define these groups, prove they are simple, and describe their automorphisms, subgroups and covering groups.

The classical groups

Algebra, by P. M. Cohn. Vol. 1. Pp xii, 321. £6·50 cloth, £2·90 paper. 1974. SBN 0 471 16430 5/16431 3 (Wiley)

https://www.math.auckland.ac.nz/~obrien/research/BHLOjscrev.pdf

A practical model for computation with matrix groups

We study incidence geometries that are thin and residually connected. These geometries generalise abstract polytopes. In this generalised setting, guided by the ideas from the polytope theory, we introduce the concept of chirality, a property of orderly asymmetry occurring frequently in nature as a natural phenomenon. The main result in this paper is that automorphism groups of regular and chiral thin residually connected geometries need to be C-groups in the regular case and \({C^+}\)-groups in the chiral case.

Highly symmetric hypertopes

We provide the first examples of words in the free group of rank 2 which are not proper powers and for which the corresponding word maps are non-surjective on an infinite family of finite non-abelian simple groups.

Some word maps that are non-surjective on infinitely many finite simple groups

We give explicit formulas for the cohomology of the Heisenberg Lie algebras over fields of finite characteristic. We use this to show that in characteristic two, unlike all other cases, the Betti numbers are unimodal.

/pdf/the-cohomology-of-the-heisenberg-lie-algebras-over-fields-of-gjnluerhxw.pdf

Sebastian Jambor

Papers

Some word maps that are non-surjective on infinitely many finite simple groups

The cohomology of the Heisenberg Lie algebras over fields of finite characteristic

Normal forms for matrices over uniserial rings of length two

C-groups of PSL(2,q) and PGL(2,q)

An L2-quotient algorithm for finitely presented groups on arbitrarily many generators