College Algebra Exam Review 140

College Algebra Exam Review 140 - 150 3 PRODUCTS OF GROUPS...

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Unformatted text preview: 150 3. PRODUCTS OF GROUPS constructed previously from the Cartesian product of A and B is the exter- nal direct product. On the other hand, if a group G has normal subgroups N and M such that N \ M D f e g and NM D G , so that G is isomorphic to the direct product N M , G is said to be the internal direct product of N and M . The distinction is more psychological than mathematical. When the groups involved are abelian and written with additive no- tation, it is common to use the terminology direct sum instead of direct product and use the notation A ˚ B instead of A B . In this book, we shall, in general, use the terminology of direct sums and the notation ˚ for abelian groups with additional structure (for example, rings, vector spaces, and modules) but we shall stay with the terminology of direct products and with the notation when speaking of abelian groups as groups . We can define the direct product of any finite number of groups in the same way as we define the direct product of two groups. As a set, the directsame way as we define the direct product of two groups....
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