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Math Help - Union of proper subgroups

  1. #1
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    Union of proper subgroups

    The following problem was given on a test of mine and I got it completely wrong. If anyone can help me with solving this problem that would be great

    Let H and K be a subgroup of G, such that H is not equal to G and K is not equal to G . Prove that H union K is not equal to G. Hint: A group cannot be the union of two proper subgroups
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  2. #2
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    Re: Union of proper subgroups

    If H=K, or one is contained in another is trivial. Let h be in H but not in K and k be in K but not in H. Then imagine h*k is in H. then h*k=h' or equivalently k=h^-1*h', which is a contradition, because it would mean k is in H. Similarly for the supposition that h*k=k'. Thus h*k is not in H or K, but is in G. QED
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  3. #3
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    Re: Union of proper subgroups

    one can also argue by contradiction. suppose G = HUK and that H,K < G.

    since K is proper, pick x in G-K. since G = HUK, x must lie in H.

    now pick ANY k in K. since x = (xk)k-1, xk cannot be in K, or else x would be also. so xk is in H.

    since H is a subgroup, x-1 is also in H, so by closure x-1(xk) = k is in H.

    thus K is a subgroup of H, so HUK = H ≠ G, contradiction.
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