= Simply transitive group action
{title2=$\forall x,y\ \exists!g:\ gx=y$}
A <group action> on a nonempty <set> $X$ is simply transitive when exactly one <group> element sends any chosen point to any other:
$$
\forall x,y\in X\quad\exists!g\in G:\quad g\cdot x=y.
$$
Equivalently it is both a <transitive group action> and a <free group action>; this is also a <regular group action>. Transitivity supplies a transport, and freeness makes it unique. Conversely, putting $y=x$ in the uniqueness condition shows that every <stabilizer subgroup> is $\{e\}$, since the <identity element> already fixes $x$. In a <transitive group action>, checking that just one <stabilizer subgroup> is trivial suffices: all the others are <conjugate subgroups>.
The <cyclic group> $C_3$ in the diagram under <transitive group action> is simply transitive. More generally, the <left regular action> of any <group> $G$ on itself is simply transitive: the equation $gx=y$ has the unique solution $g=yx^{-1}$, using the <inverse element> of $x$. This shows why the property is useful: every position has a unique label relative to a chosen starting position.
Choose a point $x_0\in X$. Its <orbit map>
$$
\phi_{x_0}:G\longrightarrow X,\qquad g\longmapsto g\cdot x_0
$$
is a <bijection> and an <equivariant map> from the <left regular action>: $\phi_{x_0}(hg)=h\cdot\phi_{x_0}(g)$. This identifies $X$ with a copy of $G$, but $X$ does not come with a preferred <identity element>. If $x_1=a\cdot x_0$, then $\phi_{x_1}(g)=\phi_{x_0}(ga)$: changing the chosen point composes this identification with right multiplication by $a$. An old label $g$ therefore becomes the new label $ga^{-1}$, which also specifies the side of multiplication for a noncommutative <group>.
The transport from $x$ to $y$ remains independent of the chosen point. Writing $x=b\cdot x_0$ and $y=c\cdot x_0$, it is $cb^{-1}$. With $x_1=a\cdot x_0$, the new labels are $ba^{-1}$ and $ca^{-1}$, and
$$
(ca^{-1})(ba^{-1})^{-1}=cb^{-1}.
$$
For a <finite group>, simple transitivity implies $|X|=|G|$. Conversely, a <transitive group action> with $|X|=|G|<\infty$ is simply transitive by the <orbit-stabilizer theorem>. Equality of <cardinalities> alone is insufficient without transitivity: the <cyclic group> $C_2$ can fix both points of a two-point <set>, giving two <group orbits> and two possible self-transports at each point.
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