= Transitive group action
{wiki}
A <group action> on a nonempty <set> $X$ is transitive when every two points lie in the same <orbit of a group action>, equivalently when for all $x,y\in X$ there is a <group> element $g$ with $g\cdot x=y$. There is exactly one <group orbit>: every position is reachable from every other. Requiring $X\ne\varnothing$ avoids calling the <group action> on the <empty set> transitive merely because the quantified statement has no points to test.<Transitive Group Actions by Keith Conrad>[Definition 1.1, printed page 1: transitivity requires a nonempty set and exactly one orbit]{ref}
Here a left <group action> of a <group> $G$ means a <function> $G\times X\to X$, written $(g,x)\mapsto g\cdot x$, satisfying
$$
e\cdot x=x,\qquad (gh)\cdot x=g\cdot(h\cdot x),
$$
where $e$ is the <identity element>. Products move the point by $h$ first, then by $g$. Transitivity says that an <orbit map> $G\to X$, $g\mapsto g\cdot x$, is <surjective> for every $x\in X$; it suffices to check one chosen point.<Group Actions by Keith Conrad>[Definition 1.3, printed page 2: the two action axioms]{ref}<Transitive Group Actions by Keith Conrad>[Theorem 3.1, printed page 2: equivalent tests for transitivity]{ref}
For example, the <symmetric group> $S_3$ acts on $X=\{1,2,3\}$ by $\sigma\cdot i=\sigma(i)$. This is a <transitive group action>, but it is not a <free group action>: the <stabilizer subgroup> of $1$ is $\{e,(23)\}$. Both the <transposition> $(12)$ and the <cyclic permutation> $(123)$ send $1$ to $2$. In contrast, the <cyclic group> $C_3=\mathbb Z/3\mathbb Z$ acts on itself by $k\cdot x=k+x\pmod3$, using <modular arithmetic>. The <group> element sending $x$ to $y$ is uniquely $k=y-x\pmod3$. These two <group actions> reach the same number of points, but only the second has unique transport between them.
\Image[/transitive-actions-symmetric-and-cyclic.png]
{title=<Transitive group actions> with and without unique transport}
{id=transitive-actions-symmetric-and-cyclic}
{description=The <symmetric group> $S_3$ on three labels has a <stabilizer subgroup> of <group order> two at each point, hence two <group> elements sending any chosen point to any other. The <cyclic group> $C_3$ acting on itself has trivial <stabilizer subgroups> and one transport per pair. Red arrows show a generating <cyclic permutation>; blue arrows show the extra <transposition> in $S_3$. Gray loops show the <identity element> in both <group actions>.}
The <stabilizer subgroup> $G_x=\{h\in G:h\cdot x=x\}$ measures the ambiguity of transport. If $g_0\cdot x=y$, then
$$
\{g\in G:g\cdot x=y\}=g_0G_x,\qquad G_y=g_0G_xg_0^{-1}.
$$
The first <set> is a left <coset>, and the second identity says that <stabilizer subgroups> in one <group orbit> are <conjugate subgroups>. Briefly, $g\cdot x=g_0\cdot x$ exactly when $g_0^{-1}g\in G_x$. Thus, for a <finite group> in a <transitive group action>, the <orbit-stabilizer theorem> gives $|G|=|X|\,|G_x|$, and there are exactly $|G_x|$ transports for each pair of points. The example counts are $6=3\cdot2$ for $S_3$ and $3=3\cdot1$ for $C_3$. This separates the number of reachable positions from the number of ways to reach each one.<Group Actions by Keith Conrad>[Theorem 3.16(b)-(c), printed pages 12-13: conjugate stabilizers, transport cosets and the orbit-stabilizer formula]{ref}
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