Page 57 - DMTH403_ABSTRACT_ALGEBRA

Abstract Algebra Richa Nandra, Lovely Professional University

Notes Unit 4: Lagrange's Theorem

CONTENTS
Objectives

Introduction
4.1 Cosets
4.2 Lagrange's Theorem
4.3 Summary
4.4 Keywords

4.5 Review Questions
4.6 Further Readings

Objectives

After studying this unit, you will be able to:
Discuss the cosets of a subgroup

Explain the partition a group into disjoint cosets of a subgroup

Prove and explain Lagrange's theorem

Introduction

In the last unit, you have studied about the subgroup and different properties of subgroups. In
this unit, you will learn the concept of cosets and also see how a subgroup can partition a group
into equivalence classes. You can use cosets to prove a very useful result about the number of
elements in a subgroup. In the present era, this elementary theorem is known as Lagrange's
theorem, though Lagrange proved it for subgroups of S only. Let us understand these concepts
with the help of examples and theorem.

4.1 Cosets

First of all we will discuss cosets. Cosets means the product of two subset of a particular group.
In a case when one of the subsets consists of single element only, we will go through a situation
i.e.,
H(x) = {hx | h H}.
where H is a subgroup of G and x  G, we will denote H{x} by Hx.

Definition: Let H be a subgroup of a group G, and let x  G. We call the set
Hx = {hx | h  H}
a right coset of H in G. The element x is a representative of Hx.

We can similarly define the left coset

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