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Math Help - Cardinality of Subgroup H n K

  1. #1
    Junior Member
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    Cardinality of Subgroup H n K

    Hi,

    Given that H \leq G and K \leq G.How do I show that \mid H \cap K\mid = gcd(\mid H\mid,\mid K\mid)?
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  2. #2
    MHF Contributor Swlabr's Avatar
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    This isn't true. For example, take G = H \times K with H \cong K. Then |(H \times e_K) \cap (e_H \times K)| = 1 but gcd(|K|, |H|) = |K| = |H|...

    It is, however, true that |H \cap K| divides gcd(|H|, |K|), and this is because it is a subgroup of both H and of K, so it's order must divide both |H| and |K|.
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