Kernel of an action; faithful, free, transitive, simply transitive actions

Intuitive definition

The kernel measures what the action "does not see" of $G$; the action is faithful when it sees everything. It is transitive when $X$ is in one piece (a single orbit), free when no non-trivial element fixes any point, simply transitive when every point can be sent to every other in exactly one way.

Formal definition

Let $G$ act on $X$ and let $\tau : G \to \mathfrak{S}(X)$ be the associated homomorphism.

• The kernel of the action is
  $$\ker \tau = \{ g \in G \mid \forall x \in X,\ g \cdot x = x \} = \bigcap_{x \in X} G_x .$$It is a normal subgroup of $G$.
• The action is faithful if $\ker \tau = \{e\}$: then $G$ identifies with a subgroup of $\mathfrak{S}(X)$.
• The action is free if all stabilizers are trivial: $g \cdot x = x \Rightarrow g = e$.
• The action is transitive if there is only one orbit: for all $x, y \in X$ there exists $g \in G$ with $y = g \cdot x$. One says $X$ is a homogeneous space under $G$.
• The action is simply transitive if it is free and transitive: for all $x, y$ there is a unique $g$ with $y = g \cdot x$. One says $X$ is a $G$-torsor (principal homogeneous space).
• The action is $k$-transitive if it is transitive on $k$-tuples of pairwise distinct elements.

Properties

• Free implies faithful; the converse fails (see example 4).
• Any action induces a faithful action of the quotient $G/\ker \tau$ on $X$.
• If the action is transitive, $X$ is in bijection with $G/G_x$ for any $x$ (orbit-stabilizer theorem); if it is simply transitive, $X$ is in bijection with $G$, but the bijection depends on the choice of a base point.

Examples

1. Free, not transitive. $\mathbb{Z}$ on $\mathbb{R}$ by translation $n \cdot x = x + n$.
2. Transitive, faithful, not free. $\mathfrak{S}_n$ on $\{1, \ldots, n\}$ for $n \geq 3$: it is even $n$-transitive, and $\mathfrak{A}_n$ is $(n-2)$-transitive.
3. Simply transitive. $G$ on itself by left translation; $\mathrm{GL}_n(K)$ on the bases of $K^n$; $\mathrm{O}_n(\mathbb{R})$ on the orthonormal bases of $\mathbb{R}^n$; the additive group $(E, +)$ of a vector space on an affine space with direction $E$ (this is the very definition of an affine space).
4. Faithful, not free. $\mathfrak{S}_3$ on the $2$-element subsets of $\{1, 2, 3\}$: a permutation fixing every pair fixes their intersections, hence every point, so the action is faithful; but $\operatorname{Stab}(\{1, 2\}) = \{ \mathrm{id}, (1\ 2) \}$.
5. Not faithful. $G$ on itself by conjugation: the kernel is the center $Z(G)$. $\mathrm{GL}_n(K)$ on the lines of $K^n$: the kernel is the group of homotheties $K^\ast I_n$, whence the faithful action of $\mathrm{PGL}_n(K)$ on projective space.
6. $G$ on $G/H$ by translation: transitive, with kernel $\bigcap_{g \in G} g H g^{-1}$, the largest normal subgroup of $G$ contained in $H$ (extended Cayley theorem).

Remarks

• Classical pitfall. Do not confuse free and faithful: free means every stabilizer is trivial, faithful means the intersection of the stabilizers is trivial.
• To prove transitivity: either exhibit explicitly $g$ sending $x$ to $y$, or count and show $|G \cdot x| = |X|$.
• To prove faithfulness: show that every $g \neq e$ moves at least one point.
• Pour prouver la fidélité : montrer que tout $g \neq e$ déplace au moins un point.