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 $xaH=aH$. 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.

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 $x$ 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.