# Name/notation for the subgroup generated by all stabilizers

Say we have a group $G$ acting on a set $X$.

I'm interested in the subgroup generated by all isotropy groups $G_x$, and looking for a designation for it.

PS1: I thought about the words 'radical' and 'residual' but apparently they don't work.

PS2: It would be nice if the notation would take into account $G$ and $X$ too.

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Invent whatever name you like and whatever notation you want. Enjoy your freedom! – Mariano Suárez-Alvarez Feb 13 '13 at 20:38

If the action is transitive, then you are talking of the normal closure of a stabilizer $G_x$.
And in general, it is the subgroup generated by the normal closures of the $G_x$, where $x$ runs through a set of representatives of the orbits. – Martin Brandenburg Feb 14 '13 at 16:38