Group Action on Subgroup of Symmetric Group

Theorem

Let $S_n$ be the symmetric group of $n$ elements.

Let $H$ be a subgroup of $S_n$.

Let $X$ be any set with $n$ elements.

Then $H$ acts on $X$ as a group of transformations on $X$.

Proof

The identity permutation takes each element of $X$ to itself, thus fulfilling Group Action Axiom $\text {GA} 2$.

The group operation in $S_n$ ensures fulfilment of Group Action Axiom $\text {GA} 1$.

$\blacksquare$

Also see

• Stabilizer in Group of Transformations

Sources

• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: The Symmetric Groups: $\S 76$
• 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $10$: The Orbit-Stabiliser Theorem: Example $10.2$