Proof that Z x Z is not a cyclic group

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Proof that Z x Z is not a cyclic group. We use a proof by contradiction.
It follows that the direct product of ANY two infinite cyclic groups is not cyclic. To see this let ~ denote isomorphism and note that if G and H are infinite cyclic groups, then G ~ Z and H ~ Z. It follows that G x H ~ Z x Z which is not cyclic, hence G x H is not cyclic either.

[@TheMathSorcerer]

[@busbusvlog]

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[@LibertyAzad]

Helps here too. Thank you!

[@priyak8943]

Determine the order of (ZxZ) / .Is the group is cyclic?

[@billy2533]

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Best wishes.

[@TheMathSorcerer]

you got it man! I know this stuff is so hard to learn!!

[@jonathangallagher3116]

The old sorcerer saves the day.... on day 76 of my math challenge! Currently working through Dan Sarancinos book and came across this problem. Cheers!

[@Mathswithsasanka]

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[@TheMathSorcerer]

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[@yanwang238]

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[@pankajkumar-hg9is]

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[@Monirul2512]

Thank you sir

[@TheMathSorcerer]

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[@boki6939]

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[@rupalimohanty9920]

Tas lot