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differential geometry w notes from teacher_Part_84

# differential geometry w notes from teacher_Part_84 - An...

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7.1. ALGEBRAIC PRELIMINARIES 167 7.1.2 Finitely Generated and Free Abelian Groups Let G be an Abelian group. Let g 1 , . . . , g r G be some elements of G and H = r k = 1 n k g k | g k G , n k Z be a set of linear combinations of g k . Then H is a subgroup of G . The elements g k are called the generators of H and H is said to be generated by g k . If a group G is generated by finitely many elements of G , then G is called a finitely generated group. The elements g 1 , . . . , g r are linearly independent if for any integer coe ffi - cients n 1 , . . . , n r the linear combination r k = 1 n k g k 0 is not equal to zero. A finitely generated group G is called a free Abelian group of rank r if it is generated by r linearly independent elements.

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Unformatted text preview: An Abelian group generated by one element is called a cyclic group . • Inﬁnite cyclic groups. • Finite cyclic group. • Theorem 7.1.3 Fundamental Theorem of Finitely Generated Abelian Groups. Let G be a ﬁnitely generated Abelian group with m generators. Then G is isomorphic to the direct sum of cyclic groups, G ± Z ⊕ ··· ⊕ Z | ±±±±±±± {z ±±±±±±± } r ⊕ Z k 1 ⊕ ··· ⊕ Z k p , where m = r + p. The number r is called the rank of G. Proof : di ﬀ geom.tex; April 12, 2006; 17:59; p. 165 168 CHAPTER 7. HOMOLOGY THEORY 1. ± di ﬀ geom.tex; April 12, 2006; 17:59; p. 166...
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