What's the notation for the intersection of stabilizer subgroups on a subset?

Question

Let $G$ acting on the the (finite) set $S$, or the (finite dimensional) space $V$.

Let $s \in S$, then the stabilizer $G_s:= \{ g \in G \ \vert \ gs = s \}$.

Let $R \subset S$, then there are two ways for getting a subgroup from $R$:

• $H_1=\{ g \in G \ \vert \ gr = r \ , \forall r \in R \}$
• $H_2=\{ g \in G \ \vert \ gR = R \}$

I guess that $H_2$ is noted $G_R$ and called the stabilizer subgroup on $R$, whereas $H_1 = \cap_{r \in R} G_r$.

Question: Is there a shorter notation for $H_1$? a short name?

Remark: I did not find that in Lang's book Algebra, chapter I section 5.

Answer 1

I think you mean $gR=R$ rather than $gR \subset R$ in the definition of $H_2$. That could be different if $R$ was infinite, and there is no reason to restrict attention to finite groups here.

Unfortunately, there appears to be no standard convention for this, and different authors use different notation. For example, Wielandt, in his standard textbook on permutation groups, uses $G_R$ to denote the pointwise stabilizer $H_1$, and of course lots of people have followed that, although personally I think that $G_R = H_2$ is more natural and intuitive. I have sometimes used $G_R$ for $H_2$ and $G_{(R)}$ for $H_1$. The most important thing is always to make your own notation clear, and make sure you understand the notation of whatever book or paper you are reading.

Another option (used by some CAS such as Magma) is to distinguish between sets and ordered sets (sequences). Then $G_R = H_2$ when $R$ is a plain set, but $G_R=H_1$ when $R$ is an ordered set.

Answer 2

Well there is no shorter notation, the only thing I can think of, if $\rho:G\rightarrow Sym(S)$ then you have $\rho_{|H_2}:H_2\rightarrow Sym(R)$ the induced action of $H_2$ on the subset $R$ then :

$$H_1:=Ker(\rho_{|H_2}) $$

That is $H_1$ is the kernel of the induced action of $H_2$ on $R$.