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Let $G$ be a group and $S$ a $G$-set with action $(g,s) \mapsto gs$. For some $s \in S$, let the stabilizer of $s$, $G_s=\{g \in G\,|\,gs=s\}$ be normal in $G$. What does this let us say about the action of $G$ on $S$?

I thought it might be interesting to look at an action of $G/G_s$ on $S$. However, something like $(gG_s,s) \mapsto gs$ isn't even well-defined in general.

Are there situations in which we can recover anything interesting?

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A stupid remark: because of the formula $G_{gs} = gG_sg^{-1}$, if the action is transitive, it factors through the quotient $G/G_s$. –  PseudoNeo Feb 14 '13 at 3:54

4 Answers 4

up vote 6 down vote accepted

What normality of the stabiliser says is exactly that every group element $g\in G$ that fixes $s$ also fixes the entire orbit $Gs$ pointwise. Conversely any $g\in G$ that fixes any element of the orbit $Gs$ will also fix $s$.

These two parts are equivalent, although the first sentence says that every conjugate of $G_s$ contains $G_s$, while the second sentence says that any conjugate of $G_s$ is contained in $G_s$. Even though a subgroup $H$ may strictly contain a conjugate ${}^gH$ of itself, if it contains all its conjugates then it must be equal to them all, in other words normal (should $H\supsetneq {}^gH$ then ${}^{g^{-1}}H\supsetneq H$, and the hypothesis excludes this).

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The stabilizer of a point $s \in S$ is normal in $G$ if and only if every element that stabilizes $s$ stabilizes the orbit $G\cdot s$ of $s$ pointwise.

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Assume the group action is transitive. Then all stabilizers are conjugate to one another. Therefore, if the stabilizer is normal, then all elements of $S$ have the same stabilizer. This is equivalent to saying that if $g$ fixes one point of $s$, then it fixes them all. So the action is regular: For every $s,t \in S$, there is a unique $g \in G$ such that $gs=t$. In particular, $|G| = |S|$.

(Even if the action is not transitive, then you can still say all of the above about the orbit of $s$, a point for which $G_s$ is normal.)

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Observing what happens when a stabilizer is a normal subgroup is one of my favorite pieces of math.

A priori, the conjugation action and normal subgroups do not seem all that important especially when you first begin studying group theory. However, then you are introduced to quotient groups, the Sylow Theorems, etc., and you see that they are indeed important. But the motivation is often lacking, and it seems like happy coincidence that so much information can be derived from these actions and subgroups.

However I believe some light is shed upon conjugation and normal subgroups when you consider group actions, stabilizers, and orbits (which are essentially the most important concepts unifying everything in group theory). The first result that hints that conjugation could be important is that if you have two elements that are in the same orbit of some group action then their stabilizers are conjugate subgroups. This result has already been mentioned.

Once you've recognized conjugation as an action, and you've named the fixed points of this action 'normal subgroups', you can consider what happens when a stabilizer happens to be a normal subgroup. Since by definition, conjugation doesn't move a normal subgroup, if you have the stabilizer of some element which is normal then all elements in the same orbit have the exact same stabilizer. In particular, this means that the only elements of the group which actually move the elements in the orbit are the cosets of the normal stabilizer. More interestingly, if you have two group elements from the same coset of that normal stabilizer, then they act on the orbit in the exact same manner.

These observations can actually motivate the construction of the quotient group. Let $G$ be a group, and let $N$ be a normal subgroup. Consider the coset space $G/N$, and let $G$ act on $G/N$ by left multiplication. We know in general that if we let $G$ act on $G/H$ for some subgroup $H$ by left multiplication, then the kernel of the action is $H$ and the action is transitive. However we get very convenient facts when we consider a normal subgroup. Namely, the stabilizer of any one coset $aN$ is $N$. This implies that $N\cdot aN=aN$ (as sets). Further since the cosets of $N$ act on $G/N$ in the exact same manner, we get that for all $b, b'\in bN$ that $b\cdot aN=b'\cdot aN$. This implies that for any two cosets $aN$ and $bN$ in $G/N$ that $aN\cdot bN=(ab)N$ (as sets). This means that the product set of any two cosets of $N$ is again a coset of $N$. This is a nontrivial relation. In fact, if $H$ is a subgroup of $G$ such that the product set of any two cosets of $H$ is also a coset of $H$, then $H$ must be normal (this result has been discussed here before). So, this property characterizes normal subgroups. These facts also allow us to see that $\cdot$ (subset product) is a group operation on $G/N$ whereas normally it's not even a binary operator on $G/H$ for a non-normal subgroup $H$.

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