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Richa Nandra, Lovely Professional University Unit 3: Subgroups

Unit 3: Subgroups Notes

CONTENTS
Objectives

Introduction
3.1 Subgroups
3.2 Properties of Subgroups
3.3 Cyclic Groups
3.4 Summary

3.5 Keywords
3.6 Review Questions
3.7 Further Readings

Objectives

After studying this unit, you will be able to:
Define subgroups

Explain the intersection, union and product of two subgroups

Describe structure and properties of cyclic groups

Introduction

In the last unit, you have studied about the algebraic structures of integers, rational numbers,
real numbers and complex numbers. You have got an idea that, not only is Z  Q  R  C, but
the operations of addition and multiplication coincide in these sets. In the present unit, you will
go through more examples of subsets of groups which are groups in their own right. Such
structures are rightfully named subgroups. We will discuss some of their properties also. We
will see some cases in which we obtain a group from a few elements of the group. In particular,
we will study cases of groups that can be built up by a single element of the group.

3.1 Subgroups

In the previous unit, you have already read the concept of group. You also noted that group (Z+),
(Q+) and (R+) are the member of a bigger group (C+) complex number. These all groups that
contained in bigger group are not just subsets but groups.
All these are examples of subgroup. Lets define subgroup.

Definition: Let (G,*) be a group. A non-empty subset H of G is called a subgroup of G if
(i) a * b  H  a, b  H, i.e., * is a binary operation on H,
(ii) (H,*) is itself a group.

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