Prove that the product of two infinite cyclic group is not cyclic
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If a group GG is cyclic, then the following assertion is clearly satisfied:
∀x,y∈G∖{1}, ∃n,m∈Z∖{0}, xn=ym.∀x,y∈G∖{1}, ∃n,m∈Z∖{0}, xn=ym.
Thus, in order to prove that the product of two infinite cyclic groups (ie., Z×ZZ×Z) is not cyclic, it is sufficient to notice that
n(1,0)=(n,0)≠(0,m)=m(0,1)n(1,0)=(n,0)≠(0,m)=m(0,1)
for every n,m∈Z∖{0}n,m∈Z∖{0}.
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∀x,y∈G∖{1}, ∃n,m∈Z∖{0}, xn=ym.∀x,y∈G∖{1}, ∃n,m∈Z∖{0}, xn=ym.
Thus, in order to prove that the product of two infinite cyclic groups (ie., Z×ZZ×Z) is not cyclic, it is sufficient to notice that
n(1,0)=(n,0)≠(0,m)=m(0,1)n(1,0)=(n,0)≠(0,m)=m(0,1)
for every n,m∈Z∖{0}n,m∈Z∖{0}.
hope u like it!!!
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