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Math Help - Factor group of an external direct product

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    Factor group of an external direct product

    Find the order of Z \oplus Z | <(2,2)>

    Now, the elements in Z \oplus Z are (0,0) and (1,1), so the order of it is 2. but I'm confuse about the second one, what is the order of that?
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  2. #2
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    Quote Originally Posted by tttcomrader View Post
    Find the order of Z \oplus Z | <(2,2)>

    Now, the elements in Z \oplus Z are (0,0) and (1,1), so the order of it is 2. but I'm confuse about the second one, what is the order of that?
    Here \mathbb{Z}\oplus \mathbb{Z} = \{... ,(-2,-2),(-1,-1),(0,0),(1,1),(2,2),...\}
    Write out \left<(2,2)\right> which means \{...,(-4,-4),(-2,-2),(0,0),(2,2),(4,4),...\}.
    Now notice that,
    \left< (2,2) \right> \mbox{ and }(1,1) + \left< (2,2)\right>.
    Form a complete list of co-sets, i.e. the even ones and the odd ones.
    Thus, this factor group is of order 2.
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