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  1. #1
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    Stuck on this question, struggling to see where to start.
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  2. #2
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    Each element of HK is of the form hk, where h\in H,\;k\in K. But there may be more than one way of representing the same element as such a product.

    So suppose that h_1k_1 = h_2k_2. Then h_2^{-1}h_1 = k_2k_1^{-1} = p say, where p\in H\cap K (because p = h_2^{-1}h_1\in H and also p = k_2k_1^{-1}\in K).

    Use that to count the number of ways that each element of HK can be expressed as a product of the form hk.
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  3. #3
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    where
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  4. #4
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    What about the counter-example?
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  5. #5
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    Quote Originally Posted by math_help View Post
    What about the counter-example?
    You could take G = S_3, with H and K the subgroups of order 2 generated by the transpositions (12) and (13) respectively.
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