1. Tate’s theorem 2 Exercises 5 2. Introduction to group schemes 5 Definition (as a functor) 6 Definition (as a group object) 6 Examples of group schemes 8 Rank and the augmentation ideal 9 Subgroup schemes, morphisms and kernels 11 Diagonalizable group schemes 13 Constant group schemes 14 Exercises 14 3. Duality and Deligne’s theorem 16 Cartier duality 16 Deligne’s theorem 19 Exercises 22 4. É...

In the preface of Finite Group Theory the author, I. Martin Isaacs, states that his principal reason for writing the book was to expose students to the beauty of the subject. Finite group theory is indeed a subject which has both beautiful theory and beautiful examples. The simplicity and elegance of the group axioms have made group theory an almost universal choice as a starting point in the t...

There is a large collection of e ective algorithms for computing information about nite soluble groups. The success in computation with these groups is primarily due to a computationally convenient representation of them by means of (special forms of) power conjugate presentations. A notable omission from this collection of algorithms is an e ective algorithm for computing the automorphism grou...

In a remarkable variety of contexts appears the modular data associated to finite groups. And yet, compared to the well-understood affine algebra modular data, the general properties of this finite group modular data has been poorly explored. In this paper we undergo such a study. We identify some senses in which the finite group data is similar to, and different from, the affine data. We also ...

Abstract A finite non-regular primitive permutation group G is extremely if a point stabiliser acts primitively on each of its nontrivial orbits. Such groups have been studied for almost century, finding various applications. The classification was recently completed by Burness and Lee, who relied an earlier soluble Mann, Praeger Seress. Unfortunately, there inaccuracy in the latter classificat...

It is shown that certain torsion groups studied by Grigorchuk and by Gupta and Sidki are conjugacy separable. 1. Introduction. In [1, 2, 3], R. I. Grigorchuk introduced and studied some remarkable new examples of finitely generated torsion groups. These groups are residually finite and amenable, and they fall into 20 isomorphism classes of p-groups for each prime p. Grigorchuk solved a problem ...

We show that soluble groups G of type Bredon-FP∞ with respect to the family of all virtually cyclic subgroups of G are always virtually cyclic. In such a group centralizers of elements are of type FP∞. We show that this implies the group is polycyclic. Another important ingredient of the proof is that a polycyclic-by-finite group with finitely many conjugacy classes of maximal virtually cyclic ...