Is the direct product $\Bbb Z \times \Bbb Z$ with operation $(n,m)+(p,q):=(n+p,m+q)$ a cyclic group? I know its not a cyclic group but how would i show this in a formal way?
2026-04-01 13:43:37.1775051017
Is the direct product $\Bbb Z \times \Bbb Z$ with operation $(n,m)+(p,q):=(n+p,m+q)$ a cyclic group?
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If it were generated by one element $(a,b)$ then you'd have to have $(1,0)=n(a,b)$ which means $nb=0$. Thus $b=0$. Similarly $n(a,b)=(0,1)$ which implies $a=0$. But $(0,0)$ does not generate the whole group.