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Math Help - Operations with cosets

  1. #1
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    Operations with cosets

    I *think* this works, but it seems to easy to work. And in my experience, if it seems to simple to be true in group theory, it usually is.

    I am supposing that x in G exists in a way that no element in G conjugates x into a subgroup H.

    Now I am considering x's action on the set of H's cosets, to say then if x fixes a coset, then for some a in G we have xaH=aH. This is where I am not sure if I can do what I am doing. I then move the a over and get a^{-1}xaH=H.

    Since H is a subgroup, 1 is in it. So a^{-1}xa1 \in H or equivalently a^{-1}xa \in H. So a^{-1} conjugates x into H, contradicting my hypothesis, so x cannot fix a coset.
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  2. #2
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    Quote Originally Posted by IthacaPride View Post
    I am supposing that x in G exists in a way that no element in G conjugates x into a subgroup H.
    I do not exactly understand what this means but this is what I think you mean. Let x be an element so that x is not conjugate to any element in H (some subgroup of G). Now let x act on the left cosets of H by left-translation. If aH is a coset fixed by x then it means x(aH) = aH \implies xa = aH \implies a^{-1} xa = h for some h\in H. But then it means x is conjugate to an element in H. Thus, a contradiction.

    What you did is correct.
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