(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

In Z, let H=<5> and K=<7>. Prove that Z=HK. Is Z the internal direct product of <5> and <7>?

2. Relevant equations

3. The attempt at a solution

Since 3(5)-2(7)=1, every integer n element of Z can be written as 3n(5)+(-2n)(7), and n is an element of HK( where group operation is addition and powers written as multiples). Therefore Z=HK. Since the intersection of H and K =<35>, then Z is not equal to H direct product K.

I think this is right but I am not sure on some of the justification of my claims. Can any one help me with the justification?

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# Homework Help: Group Theroy problem

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