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Math Help - Relatively-prime-ordered subgroups

  1. #1
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    Relatively-prime-ordered subgroups

    If H and K are subgroups of a group G and if |H| and |K| are relatively prime, prove that H \cap K = \{1\}.

    I realize that this amounts to proving that |H \cap K| = 1, since the intersection of any family of subgroups is a subgroup. However, I don't know how where to go from here. Any help would be greatly appreciated!
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    Quote Originally Posted by jstew View Post
    If H and K are subgroups of a group G and if |H| and |K| are relatively prime, prove that H \cap K = \{1\}.

    I realize that this amounts to proving that |H \cap K| = 1, since the intersection of any family of subgroups is a subgroup. However, I don't know how where to go from here. Any help would be greatly appreciated!
    Let a\in H\cap K then the order of a divides both |H| and |K|. This forces the order to be one. Thus, what does it mean?
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    Ah, so |a|=1 which implies that a={1}. But a was chosen arbitrarily, so {1} is the only element in H \cap K. Is this correct?
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    Quote Originally Posted by jstew View Post
    Ah, so |a|=1 which implies that a={1}. But a was chosen arbitrarily, so {1} is the only element in H \cap K. Is this correct?
    Yes
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    Thank you for the quick, helpful replies! You rock!
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    Quote Originally Posted by jstew View Post
    You rock!
    I know. When I was a younger kid my mother used to tell me that I am the greatest person in the world. That I know everything. And that I am the most handsome.
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