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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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