Kernel of an Action

The subgroup acting trivially on every point of the set

Kernel of an Action

Let a group action of a group $G$ on a set $X$ be given. The kernel of the action is

\ker(G\curvearrowright X) := \{g\in G : g\cdot x = x \text{ for all } x\in X\}.

This is a normal subgroup of $G$ ; it is the kernel of the associated permutation homomorphism $G\to \mathrm{Sym}(X)$ . The action is faithful precisely when this kernel is trivial.

Examples:

For the trivial action, the kernel is all of $G$ .
For the left translation action of $G$ on itself, the kernel is the identity element alone.
For conjugation of $G$ on itself, the kernel is the center $Z(G)$ (elements that commute with everything).