Page 119 - DMTH403_ABSTRACT_ALGEBRA

Abstract Algebra Richa Nandra, Lovely Professional University

Notes Unit 10: Finite Abelian Groups

CONTENTS
Objectives

Introduction
10.1 Definition
10.2 Properties
10.3 Notation
10.4 Summary

10.5 Keywords
10.6 Review Questions
10.7 Further Readings

Objectives

After studying this unit, you will be able to:
Define finite abelian group

Explain the properties of finite abelian group

Discuss the notation of finite abelian group

Introduction

A group for which the elements commute (i.e., AB = BA for all elements A and B) is called a finite
abelian group. All cyclic groups are finite abelian, but a finite abelian group is not necessarily
cyclic. All subgroups of a finite abelian group are normal. In a finite abelian group, each element
is in a conjugacy class by itself, and the character table involves powers of a single element
known as a group generator. In Mathematica, the function finite abelian group
[{n , n ...}] represents the direct product of the cyclic groups of degrees n n ...
2
1
2
1
10.1 Definition
A finite abelian group is a set, A, together with an operation “•” that combines any two elements
a and b to form another element denoted a • b. The symbol “•” is a general placeholder for a
concretely given operation. To qualify as a finite abelian group, the set and operation,
(A, •), must satisfy five requirements known as the finite abelian group axioms:

Closure

For all a, b in A, the result of the operation a • b is also in A.

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