1. 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 $\displaystyle xaH=aH$. This is where I am not sure if I can do what I am doing. I then move the a over and get $\displaystyle a^{-1}xaH=H$.

Since H is a subgroup, 1 is in it. So $\displaystyle a^{-1}xa1 \in H$ or equivalently $\displaystyle a^{-1}xa \in H$. So $\displaystyle a^{-1}$ conjugates x into H, contradicting my hypothesis, so x cannot fix a coset.

2. Originally Posted by IthacaPride
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 $\displaystyle x$ be an element so that $\displaystyle x$ is not conjugate to any element in $\displaystyle H$ (some subgroup of $\displaystyle G$). Now let $\displaystyle x$ act on the left cosets of $\displaystyle H$ by left-translation. If $\displaystyle aH$ is a coset fixed by $\displaystyle x$ then it means $\displaystyle x(aH) = aH \implies xa = aH \implies a^{-1} xa = h$ for some $\displaystyle h\in H$. But then it means $\displaystyle x$ is conjugate to an element in $\displaystyle H$. Thus, a contradiction.

What you did is correct.