Lecture 20 FUNDAMENTAL Theorem of Finitely Generated Abelian Groups (FTFGAG)

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1 Lecture 20 FUNDAMENTAL Theorem of Finitely Generated Abelian Groups (FTFGAG) Warm up: 1. Let n Find all sequences n 1 n 2... n s 2 satisfying n i 1 and n 1 n s n (where s can vary from sequence to sequence) 2. Is Z 3 Z 3 cyclic? How about Z 3 Z 2? n i

2 Proposition Let m, n P Z. 1. Z m Z n Z mn if and only if pm, nq If n p α pα k k then Z n Z p α 1 1 Z α p k. k

3 Last time: 1. A group G is finitely generated if there is a finite subset A of G such that G xay.

4 Last time: 1. A group G is finitely generated if there is a finite subset A of G such that G xay. 2. For each r P Z with r 0, let Z r Z Z Z be the direct product of r copies of the group Z, where Z 0 1. The group Z r is called the free abelian group of rank r.

5 Last time: 1. A group G is finitely generated if there is a finite subset A of G such that G xay. 2. For each r P Z with r 0, let Z r Z Z Z be the direct product of r copies of the group Z, where Z 0 1. The group Z r is called the free abelian group of rank r. Theorem (FUNDAMENTAL Theorem of Finitely Generated Abelian Groups) If G is a finitely generated abelian group, then G Z r Z n1 Z n2 Z ns for some integers r, n 1 n s 2 such that r 0 and ni. Moreover, this expression is unique. n i 1

6 Theorem (FTFGAG) If G is a finitely generated abelian group, then G Z r Z n1 Z n2 Z ns for some integers r, n 1 n s 2 such that r 0 and ni. Moreover, this expression is unique. n i 1 The integer r is called the free rank or Betti number of G and the integers n 1, n 2,..., n s are called the invariant factors of G. The description is called the invariant factor decomposition of G. If G is finite, we call s the rank, and G is of type pn 1,..., n s q.

7 Theorem (FTFGAG) If G is a finitely generated abelian group, then G Z r Z n1 Z n2 Z ns for some integers r, n 1 n s 2 such that r 0 and ni. Moreover, this expression is unique. n i 1 The integer r is called the free rank or Betti number of G and the integers n 1, n 2,..., n s are called the invariant factors of G. The description is called the invariant factor decomposition of G. If G is finite, we call s the rank, and G is of type pn 1,..., n s q. Proposition Every prime divisor of n must divide the first invariant factor n 1.

8 Theorem (FTFGAG) If G is a finitely generated abelian group, then G Z r Z n1 Z n2 Z ns for some integers r, n 1 n s 2 such that r 0 and ni. Moreover, this expression is unique. n i 1 The integer r is called the free rank or Betti number of G and the integers n 1, n 2,..., n s are called the invariant factors of G. The description is called the invariant factor decomposition of G. If G is finite, we call s the rank, and G is of type pn 1,..., n s q. Proposition Every prime divisor of n must divide the first invariant factor n 1. Corollary If n is the product of distinct primes, then up to isomorphism the only abelian group of order n is Z n, the cyclic group of order n.

9 Theorem (Primary decomposition theorem) Let G be an abelian group of order n 1 where n factors into primes as n p α pα k. Then k G A 1 A 2 A k with A i p α i i.

10 Theorem (Primary decomposition theorem) Let G be an abelian group of order n 1 where n factors into primes as n p α pα k. Then k G A 1 A 2 A k with A i p α i i. Moreover, for each A P ta 1,..., A k u with A p α, A Z p β 1 Z p β t for some partition β 1 β t 1 of α.

11 Theorem (Primary decomposition theorem) Let G be an abelian group of order n 1 where n factors into primes as n p α pα k. Then k G A 1 A 2 A k with A i p α i i. Moreover, for each A P ta 1,..., A k u with A p α, A Z p β 1 Z p β t for some partition β 1 β t 1 of α. Finally, these decompositions are unique.

12 Theorem (Primary decomposition theorem) Let G be an abelian group of order n 1 where n factors into primes as n p α pα k. Then k G A 1 A 2 A k with A i p α i i. Moreover, for each A P ta 1,..., A k u with A p α, A Z p β 1 Z p β t for some partition β 1 β t 1 of α. Finally, these decompositions are unique. The integers p β j are called the elementary divisors of G. This description of G is called its elementary divisor decomposition.

13 G A 1 A 2 A k with A i p α i i. Note: A i is the unique Sylow p i -subgroup of G, so the primary decomposition theorem really says Every finite abelian group is isomorphic to a direct product of its Sylow p-subgroups.

14 PART II: RING THEORY 1. A ring R is a set together with two binary operations and such that 1.1 pr, q is an abelian group, 1.2 is associative: pa bq c a pb cq, 1.3 the distributive laws hold for R pa bq c pa cq pb cq and a pb cq pa bq pa cq.

15 PART II: RING THEORY 1. A ring R is a set together with two binary operations and such that 1.1 pr, q is an abelian group, 1.2 is associative: pa bq c a pb cq, 1.3 the distributive laws hold for R pa bq c pa cq pb cq and a pb cq pa bq pa cq. 2. The ring R is commutative if is commutative.

16 PART II: RING THEORY 1. A ring R is a set together with two binary operations and such that 1.1 pr, q is an abelian group, 1.2 is associative: pa bq c a pb cq, 1.3 the distributive laws hold for R pa bq c pa cq pb cq and a pb cq pa bq pa cq. 2. The ring R is commutative if is commutative. 3. The ring R is said to have an identity if there is an element 1 P R with 1 a a 1 a for all a P R.