Last edited by Tozil
Thursday, May 14, 2020 | History

3 edition of Finitely Generated Abelian Groups and Similarity of Matrices over a Field found in the catalog.

Finitely Generated Abelian Groups and Similarity of Matrices over a Field

by Christopher Norman

  • 46 Want to read
  • 35 Currently reading

Published by Springer London in London .
Written in English

    Subjects:
  • Group theory,
  • Matrix Theory Linear and Multilinear Algebras,
  • Field theory (Physics),
  • Group Theory and Generalizations,
  • Field Theory and Polynomials,
  • Algorithms,
  • Mathematics,
  • Matrices

  • Edition Notes

    Statementby Christopher Norman
    SeriesSpringer Undergraduate Mathematics Series
    ContributionsSpringerLink (Online service)
    The Physical Object
    Format[electronic resource] /
    ID Numbers
    Open LibraryOL27038860M
    ISBN 109781447127307

    group generated by matrices of finite order. Ask Question Asked 2 years, 11 months ago. Active 2 years, Describe the finite order integer matrices over complex field. 0. A kind of pumping lemma for finitely generated groups. 6. Abstract Algebra Theory and Applications. This text is intended for a one- or two-semester undergraduate course in abstract algebra. Topics covered includes: The Integers, Groups, Cyclic Groups, Permutation Groups, Cosets and Lagrange’s Theorem, Algebraic Coding Theory, Isomorphisms, Normal Subgroups and Factor Groups, Matrix Groups and Symmetry, The Sylow Theorems, Rings, Polynomials.

    Corollary: A finitely generated abelian group is free if and only if it is torsion-free, that is, it contains no element of finite order other than the identity. The number \(r\) is called the rank of \(A\). The Springer Undergraduate Mathematics book series by multiple authors includes books Derivative Pricing in Discrete Time, Finitely Generated Abelian Groups and Similarity of Matrices Over a Field, Worlds Out of Nothing: A Course in the History of Geometry in the 19th Century, and several more.

    Abelian group explained. In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, these are the groups that obey the axiom of n groups generalize the arithmetic of addition of integers. Decomposition of Finitely Generated Z-Modules. Part 2: Similarity of Square Matrices over a Field: The Polynomial Ring F[x] and Matrices over F[x]- F[x] Modules: Similarity of t xt Matrices over a.


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Finitely Generated Abelian Groups and Similarity of Matrices over a Field by Christopher Norman Download PDF EPUB FB2

The theory of finitely generated abelian groups is introduced in an understandable and concrete way. The analogous theory of similarity of square matrices over a field, including the Jordan form, is explained step by step.

Manipulative techniques are stressed to get students ‘on board’.Brand: Springer-Verlag London. Starting with matrices over the integers, Part 1 of this book provides a measured introduction to such groups: two finitely generated abelian groups are isomorphic if and only if their invariant factor sequences are identical.

The analogous theory of matrix similarity over a field Cited by: 9. The theory of finitely generated abelian groups is introduced in an understandable and concrete way The analogous theory of similarity of square matrices over a field, including the Jordan form, is explained step by step Manipulative techniques are stressed to get students ‘on board’ At first sight, finitely generated abelian groups and canonical forms of matrices appear to have little in common.

At first sight, finitely generated abelian groups and canonical forms of matrices appear to have little in common.

However, reduction to Smith normal form, named after its originator inis a matrix version of the Euclidean algorithm Author: Christopher Norman. springer, At first sight, finitely generated abelian groups and canonical forms of matrices appear to have little in common.

However, reduction to Smith normal form, named after its originator inis a matrix version of the Euclidean algorithm and is exactly what the theory requires in both cases. At first sight, finitely generated abelian groups and canonical forms of matrices appear to have little in common.

However, reduction to Smith normal form, named after its originator inis a matrix version of the Euclidean algorithm and is exactly what the theory requires in both cases. - Buy Finitely Generated Abelian Groups and Similarity of Matrices over a Field (Springer Undergraduate Mathematics Series) book online at best prices in India on Read Finitely Generated Abelian Groups and Similarity of Matrices over a Field (Springer Undergraduate Mathematics Series) book reviews & author details and more at Free delivery on Author: Christopher Norman.

Finitely generated abelian groups and similarity of matrices over a field 1. 2 Basic Theory of Additive Abelian GroupsIn this chapter we discuss cyclic groups, the quotient group construction, the directsum construction and the first isomorphism theorem, in the context of additive abeliangroups; we also discuss free modules.

Classi cation of Finitely Generated Abelian Groups. The proof given below uses vector space techniques (Smith Normal Form) and gener- alizes from abelian groups to \modules over PIDs" (essentially generalized vector spaces).

Smith Normal Form is a reduced form similar to the row reduced matrices encountered in elementary linear Size: 84KB. That is, every finitely generated abelian group is isomorphic to a group of the form where n ≥ 0 is the rank, and the numbers q1,qt are powers of (not necessarily distinct) prime numbers.

In particular, G is finite if and only if n = 0. The values of n, q1,qt are (up to rearranging the indices). Starting with matrices over the integers, Part 1 of this book provides a measured introduction to such groups: two finitely generated abelian groups are isomorphic if and only if their invariant factor sequences are : Springer London.

Abstract. The matrix A of a linear mapping α of a t-dimensional vector space relative to a basis, similarity classes, the characteristic polynomial χ A (x).The F[x]-modules M(α) and M(A).Isomorphic modules M(A) are determined by similar matrices A and conversely. Order of a module element. Submodules of M(A).Direct sum of square matrices.

General decomposition of M(A).Author: Christopher Norman. Starting with matrices over the integers, Part 1 of this book provides a measured introduction to such groups: two finitely generated abelian groups are isomorphic if and only if Brand: Springer London.

This book provides an introduction to the decomposition of finitely generated abelian groups and canonical forms of matrices, and explores the analogous theory of matrix similarity over a field. Includes numerous worked examples and exercises with solutions.

Finitely Generated Abelian Groups Finitely generated abelian groups arise all over algebraic number theory. For example, they will appear in this book as class groups, unit groups, and the underlying additive groups of rings of integers, and as Mordell-Weil groups of elliptic curves.

Section II Direct Products and Finitely Generated Abelian Groups Note. In the previous section, we took given groups and explored the existence of subgroups. In this section, we introduce a process to build new (bigger) groups from known groups.

This process will allow us to classify all finite abelian groups. Definition File Size: 62KB. Starting with matrices over the integers, Part1 of this book provides a measured introduction to such groups: two finitely generated abelian groups are isomorphic if and only if.

Over a Noetherian ring the concepts of finitely generated, finitely presented and coherent modules coincide. A finitely generated module over a field is simply a finite-dimensional vector space, and a finitely generated module over the integers is simply a finitely generated abelian group.

Every finitely generated abelian group is a direct sum of cyclic groups, that is, of the form. The first summands are the torsion subgroup, and the last one is the free subgroup. Ordinarily I’d be glad to stop here, but the theorem is usually stated with a little more detail, so I’ll go on.

The kernel is itself finitely generated (this takes a little argument in and of itself; note that adding a new generator to a subgroup of free abelian group either increases dimension after tensoring with $\mathbb{Q}$ or descreases the size of the torsion of the quotient), so our group is the cokernel of a map between finite rank free groups.

Similarity of Matrices over a Field. Christopher Norman formerly Senior Lecturer in Mathematics Part I Finitely Generated Abelian Groups 1 Matrices with Integer Entries: The Smith Normal Form Springer Undergraduate Mathematics Series.

Theorem. Let be a finitely generated abelian group. Then, for some and. We call the torsion part of. On the other hand, infinite abelian groups are far from classified.

(See here.) Classical examples would be the cyclic groups or the Klein four-group. Working with abelian groups might lead to the feeling that there must be some sort of free.Structure of Finitely Generated Abelian Groups by T.

J. Pepper Ap The report of the investigation undertaken as a Senior Thesis, to carry one course of credit in the Department of Mathematics _____ _____ Michael T. Orr David Yuen, Chairperson Krebs Provost and Author: Turner J. Pepper.