Use the Krull-Schmidt Theorem to prove Theorems 2.2 and 2.6 (iii) for finite abelian groups. Data From

Question:

Use the Krull-Schmidt Theorem to prove Theorems 2.2 and 2.6 (iii) for finite abelian groups.

Data From Theorem 2.2

Every finitely generated abe/ian group G is (isomorphic to) a finite direct sum of cyclic groups, each of which is either infinite or of order a power of a prime.

SKETCH OF PROOF.

The theorem is an immediate consequence of Theorem 2.1 and the following lemma. Another proof is sketched in Exercise 4.

Data From Theorem 2.6

Let G he a finitely generated abelian group.(i) There is a unique nonnegative integer s such that the number of infinite cyclic summands in any decomposition of G as a direct sum of cyclic groups is precisely s;(ii) either G is free abelian or there is a unique list of (not necessarily distinct) positive integers m₁, ... , m_t such that m₁ > 1, m₁| m₂|· · · |m_t and

with F free abelian;(iii) either G is free abelian or there is a list of positive integerl P₁⁸¹ , ••• , P_k^8k, which is unique except for the order of its members, such that p₁, ... , P_k are (not necessarily distinct) primes, s₁, ... , s_k are (not necessarily distinct) positive integers and

with F free abelian.

PROOF.

(i) Any decomposition of Gas a direct sum of cyclic groups (and there is at least one by Theorem 2 .1) yields an isomorphism G ≅ H⊕F, where H is a direct sum of finite cyclic groups (possibly 0) and Fis a free abelian group whose rank is precisely the number s of infinite cyclic summands in the decomposition. If _L : H → H⊕F is the canonical injection (h |→ (h,0)), then clearly ,_L(H) is the torsion subgroup of H⊕F. By Lemma 2.5, G_t≅_L(H) under the isomorphism G ≅ H⊕F. Consequently by Corollary 1.5.8, G/G_t≅ (F⊕H)/_L(H) ≅ F. Therefore, any decomposition of G leads to the conclusion that G/G_t, is a free abelian group whose rank is the numbers of infinite cyclic summands in the decomposition. Since G/G_t, does not depend on the particular decomposition and the rank of G/G_t is an invariant by Theorem 1.2, s is uniquely determined.(iii) Suppose G has two decompositions, say