Group Action of Symmetric Group Acts Transitively

Theorem

Let $S$ be a set.

Let $\struct {\map \Gamma S, \circ}$ be the symmetric group on $S$.

Let $*: \map \Gamma S \times S \to S$ be the group action defined as:

$\forall \pi \in \map \Gamma S, \forall s \in S: \pi * s = \map \pi s$

Then $*$ is a transitive group action.

In other words, $\struct {\map \Gamma S, \circ}$ acts transitively on $S$ by $*$.

Proof

By Group Action of Symmetric Group, $*: \map \Gamma S \times S \to S$ is indeed a group action

Let $s, t \in S$.

As $\map \Gamma S$ is the symmetric group on $S$, there exists a permutation $\pi \in \map \Gamma S$ such that:

$\map \pi s = t$

This holds for any $s, t \in S$.

Thus:

$\forall t \in S: t \in \Orb s$

and so $S$ consists of a single orbit.

Hence the result by definition of transitive group action.

$\blacksquare$

Sources

• 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 3.3$: Group actions and coset decompositions: Examples of group actions: $\text{(ii)}$