Stabilizer of Polynomial

Theorem

Let $n \in \Z: n > 0$.

Let $\map f {x_1, x_2, \ldots, x_n}$ be a polynomial in $n$ variables $x_1, x_2, \ldots, x_n$.

Let $S_n$ denote the symmetric group on $n$ letters.

Let $\pi, \rho \in S_n$.

Let the group action $\pi * f$ be defined as the permutation on the polynomial $f$ by $\pi$.

Then the stabilizer of $f$ is the set of permutations on $n$ letters which fix $f$.

Proof

Follows directly from the definition of the stabilizer of $f$.

$\blacksquare$

Sources

• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): Chapter $6$: Cosets: Exercise $10$
• 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $10$: The Orbit-Stabiliser Theorem: Example $10.10$