Nilpotent group

In mathematics, specifically group theory, a nilpotent group G is a group that has an upper central series that terminates with G. Equivalently, its central series is of finite length or its lower central series terminates with {1}. Intuitively, a nilpotent group is a group that is "almost abelian". This idea is motivated by the fact that nilpotent groups are solvable, and for finite nilpotent groups, two elements having relatively prime orders must commute. It is also true that finite nilpotent groups are supersolvable. The concept is credited to work in the 1930s by Russian mathematician Sergei Chernikov. Nilpotent groups arise in Galois theory, as well as in the classification of groups. They also appear prominently in the classification of Lie groups. Analogous terms are used for Lie algebras (using the Lie bracket) including nilpotent, lower central series, and upper central series. Definition The definition uses the idea of a central series for a group. The followi

About this result

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications

Related people

Related units

Related concepts

Related courses

Related lectures

Summary

Official source

About this result

Related publications (12)

Related publications

Related people (1)

Related people

Related units (1)

Related units

Related concepts (25)

Related concepts

Related courses (3)

Related courses

Related lectures (2)

Related lectures

Varieties of modules for Z/2Z x Z/2Z

Let k be an algebraically closed field of characteristic 2. We prove that the restricted nilpotent commuting variety C, that is the set of pairs of (n x n)-matrices (A, B) such that A(2) = B-2 = [A, B] = 0, is equidimensional. C can be identified with the 'variety of n-dimensional modules' for Z/2Z x Z/2Z, or equivalently, for k[X, Y]/(X-2, Y-2). On the other hand, we provide an example showing that the restricted nilpotent commuting variety is not equidimensional for fields of characteristic > 2. We also prove that if e(2) = 0 then the set of elements of the centralizer of e whose square is zero is equidimensional. Finally, we express each irreducible component of C as a direct sum of indecomposable components of varieties of Z/2Z x Z/2Z-modules. (c) 2007 Elsevier Inc. All rights reserved.

Elsevier2007

Equivalences between blocks of p-local Mackey algebras

Baptiste Thierry Pierre Rognerud

Let G be a finite group and (K, O, k) be a p-modular system. Let R = O or k. There is a bijection between the blocks of the group algebra and the blocks of the so-called p-local Mackey algebra mu(1)(R)(G). Let b be a block of RG with abelian defect group D. Let b' be its Brauer correspondant in N-G(D). It is conjectured by Broue that the blocks RGb and RNG(D)b' are derived equivalent. Here we look at equivalences between the corresponding blocks of p-local Mackey algebras. We prove that an analogue of the Broue's conjecture is true for the p-local Mackey algebras in the following cases: for the principal blocks of p-nilpotent groups and for blocks with defect 1.

Elsevier2015

We determine the torsion subgroup of the group of endotrivial modules for a finite solvable group in characteristic p. We also prove that our result would hold for p-solvable groups, provided a conjecture can be proved about the case of p-nilpotent groups.

2011

Group theory

In abstract algebra, group theory studies the algebraic structures known as groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fi

Solvable group

In mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group that can be constructed from abelian groups using extensions. Equivalently, a solvable g

Finite group

In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just

MATH-215: Rings and fields

C'est un cours introductoire dans la théorie d'anneau et de corps.

MATH-679: Group schemes

This is a course about group schemes, with an emphasis on structural theorems for algebraic groups (e.g. Barsotti--Chevalley's theorem). All the basics will be covered towards the proof of such theorem, with an estress on the modern presentation using scheme theory and modern algebraic geometry.

MATH-735: Topics in geometric group theory

The goal of this course/seminar is to introduce the students to some contemporary aspects of geometric group theory. Emphasis will be put on Artin's Braid groups and Thompson's groups.