The Stabilizer of the coset for the action of G on $G/H$ by left multiplication.

Question

Let $H$ be a subgroup of $G$. What is the stabilizer of the coset $aH$ for the action of $G$ on $X=G/H$ by left multiplication?

So, I think I've done this one correctly: The Stabilizer is of the form: $G_{aH} = \{g \in G | gaH=aH \} \implies g \in aH \implies g=ah$ for some $h \in H \implies G_{aH} = aH$. Have I missed something?

Answer 1

Not quite. Especially, your solution $aH$ fails to be a group in general. $$gaH=aH\iff a^{-1}gaH= H\iff a^{-1}ga\in H\iff g\in aHa^{-1}$$

Alternatively: "Clearly" the stabilizer of $H$ is $H$. And for any action with $a\cdot x_1=x_2$ we have $G_{x_2}=aG_{x_1}a^{-1}$

Answer 2

You have missed something. You might check through an example.

Consider $G = \mathbb{Z}$ the integers, and $H = 5\mathbb{Z}$ the multiples of $5$. These are both groups under addition, so we'll write cosets additively. Then $G/H = \mathbb{Z}/5\mathbb{Z}$ is the additive group of $5$ elements.

Let's consider the coset $a+ H = 1 + 5\mathbb{Z}$. You've argued that the stabilizer of $1 + 5\mathbb{Z}$ is those elements of the form $a + H$, including $1 + 5\mathbb{Z}$ itself. But an element of the coset $1 + 5\mathbb{Z}$ sends $1 + 5\mathbb{Z}$ to $2 + 5\mathbb{Z}$, so it doesn't stabilize it!

Answer 3

$gaH=aH$ if and only if $ga$ is in $aH$ that is if $ga=ah$ for some $h\in H$. That is if $g=aha^{-1}\iff g\in aHa^{-1}$