Definition:Transitive Group Action

Definition

Let $G$ be a group.

Let $S$ be a set.

Let $*: G \times S \to S$ be a group action.

The group action is transitive if and only if for any $x, y \in S$ there exists $g \in G$ such that $g * x = y$.

That is, if and only if for all $x \in S$:

$\Orb x = S$

where $\Orb x$ denotes the orbit of $x \in S$ under $*$.

$n$-transitive Action

Let $n\geq1$ be a natural number.

The group action is $n$-transitive if and only if for any two ordered $n$-tuples $(x_1, \ldots, x_n)$ and $(y_1, \ldots, y_n)$ of pairwise distinct elements of $S$, there exists $g\in G$ such that:

$\forall i\in \{1, \ldots, n\} : g * x_i = y_i$

Also see

• Results about transitive group actions can be found here.

Sources

• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 6.5$. Orbits
• 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 3.3$: Group actions and coset decompositions