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Richa Nandra, Lovely Professional University Unit 9: Direct Products
Unit 9: Direct Products Notes
CONTENTS
Objectives
Introduction
9.1 Direct Product of Groups
9.1.1 External Direct Product
9.1.2 Internal Direct Product
9.2 Sylow Theorems
9.3 Groups of Order 1 to 10
9.4 Summary
9.5 Keywords
9.6 Review Questions
9.7 Further Readings
Objectives
After studying this unit, you will be able to:
Discuss direct product of groups
State Sylow theorem
Explain groups of order 1 to 10.
Introduction
In the last unit, we have studied about permutation group. This unit will provide you the
information related to 15 finite groups and direct products. Let us understand all these one by
one.
9.1 Direct Product of Groups
In this section, we will discuss a very important method of constructing new groups. We will
first see how two groups can be combined to form a third group. Then we will see how two
subgroups of a group can be combined to form another subgroup.
9.1.1 External Direct Product
In this sub-section we will construct a new group from two or more groups that we already
have.
Let (G , * ) and ((G , * ) be two groups. Consider their Cartesian product G = G × G = {(x, y) |
2
2
2
1
1
1
x G , y G }.
1
2
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