**Are you an EPFL student looking for a semester project?**

Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.

Concept# P-group

Summary

In mathematics, specifically group theory, given a prime number p, a p-group is a group in which the order of every element is a power of p. That is, for each element g of a p-group G, there exists a nonnegative integer n such that the product of pn copies of g, and not fewer, is equal to the identity element. The orders of different elements may be different powers of p.
Abelian p-groups are also called p-primary or simply primary.
A finite group is a p-group if and only if its order (the number of its elements) is a power of p. Given a finite group G, the Sylow theorems guarantee the existence of a subgroup of G of order pn for every prime power pn that divides the order of G.
Every finite p-group is nilpotent.
The remainder of this article deals with finite p-groups. For an example of an infinite abelian p-group, see Prüfer group, and for an example of an infinite simple p-group, see Tarski monster group.
Every p-group is periodic since by definition every element has finite order.
If p is prime and G is a group of order pk, then G has a normal subgroup of order pm for every 1 ≤ m ≤ k. This follows by induction, using Cauchy's theorem and the Correspondence Theorem for groups. A proof sketch is as follows: because the center Z of G is non-trivial (see below), according to Cauchy's theorem Z has a subgroup H of order p. Being central in G, H is necessarily normal in G. We may now apply the inductive hypothesis to G/H, and the result follows from the Correspondence Theorem.
One of the first standard results using the class equation is that the center of a non-trivial finite p-group cannot be the trivial subgroup.
This forms the basis for many inductive methods in p-groups.
For instance, the normalizer N of a proper subgroup H of a finite p-group G properly contains H, because for any counterexample with H = N, the center Z is contained in N, and so also in H, but then there is a smaller example H/Z whose normalizer in G/Z is N/Z = H/Z, creating an infinite descent. As a corollary, every finite p-group is nilpotent.

Official source

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 (16)

Related people (3)

Related concepts (47)

Related courses (56)

Related lectures (519)

Related MOOCs (2)

P-group

In mathematics, specifically group theory, given a prime number p, a p-group is a group in which the order of every element is a power of p. That is, for each element g of a p-group G, there exists a nonnegative integer n such that the product of pn copies of g, and not fewer, is equal to the identity element. The orders of different elements may be different powers of p. Abelian p-groups are also called p-primary or simply primary. A finite group is a p-group if and only if its order (the number of its elements) is a power of p.

Index of a subgroup

In mathematics, specifically group theory, the index of a subgroup H in a group G is the number of left cosets of H in G, or equivalently, the number of right cosets of H in G. The index is denoted or or . Because G is the disjoint union of the left cosets and because each left coset has the same size as H, the index is related to the orders of the two groups by the formula (interpret the quantities as cardinal numbers if some of them are infinite). Thus the index measures the "relative sizes" of G and H.

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 a finite number of structure-preserving transformations. Important examples of finite groups include cyclic groups and permutation groups. The study of finite groups has been an integral part of group theory since it arose in the 19th century.

MATH-506: Topology IV.b - cohomology rings

Singular cohomology is defined by dualizing the singular chain complex for spaces. We will study its basic properties, see how it acquires a multiplicative structure and becomes a graded commutative a

PHYS-431: Quantum field theory I

The goal of the course is to introduce relativistic quantum field theory as the conceptual and mathematical framework describing fundamental interactions.

PHYS-314: Quantum physics II

L'objectif de ce cours est de familiariser l'étudiant avec les concepts, les méthodes et les conséquences de la physique quantique. En particulier, le moment cinétique, la théorie de perturbation, les

Introduction to Geographic Information Systems (part 1)

Organisé en deux parties, ce cours présente les bases théoriques et pratiques des systèmes d’information géographique, ne nécessitant pas de connaissances préalables en informatique. En suivant cette

Geographical Information Systems 1

Organisé en deux parties, ce cours présente les bases théoriques et pratiques des systèmes d’information géographique, ne nécessitant pas de connaissances préalables en informatique. En suivant cette

Moodle Environment

Explores the support services and environment of Moodle at EPFL, including group modes, admin tools, course management, and user interactions.

Torsion and Divisibility in Group Theory

Explores the relationship between p-torsion and p-divisibility in group theory, highlighting implications of p-divisibility in exact sequences of abelian groups.

Identical Particles Systems: Bosons

Covers systems with identical particles, focusing on bosons and their properties.

Jacques Thévenaz, Caroline Lassueur

For the group of endo-permutation modules of a finite p-group, there is a surjective reduction homomorphism from a complete discrete valuation ring of characteristic 0 to its residue field of characte

2019The aim of this paper is to construct an equivalent of the Dade group of a p-group for an arbitrary finite group G, whose elements are equivalences classes of endo-p-permutation modules. To achieve th

This dissertation is concerned with modular representation theory of finite groups, and more precisely, with the study of classes of representations, which we shall term relative endotrivial modules.