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Math Help - product of groups

  1. #1
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    product of groups

    Prove that the product of two infinite cyclic groups is not inifinite cyclic.

    could anyone help me to prove this ?? please?
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  2. #2
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    Quote Originally Posted by jin_nzzang View Post
    Prove that the product of two infinite cyclic groups is not inifinite cyclic.

    could anyone help me to prove this ?? please?
    An infinite cyclic group is isomorphic to a group of integers \mathbb{Z} and the product of two infinite cyclic group is isomorphic to \mathbb{Z} \times \mathbb{Z}. First, Z and Z \timex Z are not isomorphic. The latter is not generated by a single element. If it were generated by a single element, it has the cardinality of a set of natural numbers, but it is not.
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  3. #3
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    Quote Originally Posted by algtop View Post
    An infinite cyclic group is isomorphic to a group of integers \mathbb{Z} and the product of two infinite cyclic group is isomorphic to \mathbb{Z} \times \mathbb{Z}. First, Z and Z \timex Z are not isomorphic. The latter is not generated by a single element. If it were generated by a single element, it has the cardinality of a set of natural numbers, but it is not.
    But \mathbb{Z} \times \mathbb{Z} does have the same cardinality as the set of natural integers!

    Nevertheless, the fact that \mathbb{Z} \times \mathbb{Z} can't be generated by a single element is true.
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  4. #4
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    Quote Originally Posted by Taluivren View Post
    But \mathbb{Z} \times \mathbb{Z} does have the same cardinality as the set of natural integers!

    Nevertheless, the fact that \mathbb{Z} \times \mathbb{Z} can't be generated by a single element is true.
    Right, a strange diagonal argument popped up in my mind when I saw this problem. I mixed that up with \{0,1\}^\omega.

    A finite product of countable sets is countable.

    For instance, there is an injective map from \mathbb{Z}_+ \times \mathbb{Z}_+ to \mathbb{Z}_+ such that f:\mathbb{Z}_+ \times \mathbb{Z}_+ \rightarrow \mathbb{Z}_+ by the equation f(n,m) = 2^n3^m. If a set has an injective map to \mathbb{Z}_+, it is countable.
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