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Thread: Infinite external direct product of groups

  1. #1 Infinite external direct product of groups 
    Moderator Moderator AlexP's Avatar
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    Jul 2006
    Just something that I thought of and am curious about... If we take a countably infinite external direct product of the group , a prime, does every non-identity element still have order ?

    In case 'external direct product' is at all ambiguous, I mean that the elements of our new group would be of the form where and multiplication is done component-wise.

    I know that in general the order of such an element with the direct product taken finitely many times is where denotes the order of . Since in this case the lcm always equals (for non-identity elements) it seems that it would work, but perhaps there's a subtlety involved with the infiniteness that I'm not seeing, and that's why I'm curious. What if we use , not prime?

    Can this somehow be done uncountably infinite times?

    "There is a kind of lazy pleasure in useless and out-of-the-way erudition." -Jorge Luis Borges
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