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Math Help - Double cosets

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    Double cosets

    If G is a finite group and H, K are subgroups of G, then how many H\G/K double cosets are there?
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  2. #2
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    Quote Originally Posted by dimuk View Post

    If G is a finite group and H, K are subgroups of G, then how many H\G/K double cosets are there?
    this is a good question! the only formula for the number of double cosets of H, K i know is in terms of the conjugacy classes of G, that is if \mathcal{C}_j, \ 1 \leq j \leq m are the conjugacy classes of G,

    then the number of double cosets of H, K is equal to: \ell=\frac{|G|}{|H||K|}\sum_{j=1}^m \frac{|\mathcal{C}_j \cap H||\mathcal{C}_j \cap K|}{|\mathcal{C}_j|}. you should check this formula for the special case when H or K is normal. note that if, for example

    K is normal, then \forall g \in G: \ HgK=HKg. thus: |HgK|=|HK|=\frac{|H||K|}{|H \cap K|}.
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