Abstract Algebra Theory and Applications - Computer Science ...

146 CHAPTER 8 ISOMORPHISMSTherefore, no (a, b) can generate all of Z m × Z n .The converse follows directly from Theorem 8.8 since lcm(m, n) = mn ifand only if gcd(m, n) = 1.□Corollary 8.11 Let n 1 , . . . , n k be positive integers. Thenk∏Z ni∼ = Zn1···n ki=1if and only if gcd(n i , n j ) = 1 for i ≠ j.Corollary 8.12 Ifm = p e 11 · · · pe kk,where the p i s are distinct primes, thenZ m∼ = Zp e 11× · · · × Z pe kk.Proof. Since the greatest common divisor of p e iiproof follows from Corollary 8.11.and p e jjis 1 for i ≠ j, the□In Chapter 11, we will prove that all finite abelian groups are isomorphicto direct products of the formZ pe 11× · · · × Z pe kkwhere p 1 , . . . , p k are (not necessarily distinct) primes.Internal Direct ProductsThe external direct product of two groups builds a large group out of twosmaller groups. We would like to be able to reverse this process and convenientlybreak down a group into its direct product components; that is,we would like to be able to say when a group is isomorphic to the directproduct of two of its subgroups.Let G be a group with subgroups H and K satisfying the followingconditions.• G = HK = {hk : h ∈ H, k ∈ K};• H ∩ K = {e};• hk = kh for all k ∈ K and h ∈ H.