Stabilizer of Cartesian Product of Group Actions

Theorem

Let $\struct {G, \circ}$ be a group.

Let $S$ and $T$ be sets.

Let $*_S: G \times S \to S$ and $*_T: G \times T \to T$ be group actions.

Let the group action $*: G \times \paren {S \times T} \to S \times T$ be defined as:

$\forall \tuple {g, \tuple {s, t} } \in G \times \paren {S \times T}: g * \tuple {s, t} = \tuple {g *_S s, g *_T t}$

Then the stabilizer of $\tuple {s, t} \in S \times T$ is given by:

$\Stab {s, t} = \Stab s \cap \Stab t$

where $\Stab s$ and $\Stab t$ are the stabilizers of $s$ and $t$ under $*_S$ and $*_T$ respectively.

Proof

By definition, the stabilizer of an element $x$ of $S$ is defined as:

$\Stab x := \set {g \in G: g * x = x}$

where $*$ denotes the group action.

So:

\(\ds \Stab {s, t}\)

\(=\)

\(\ds \set {g \in G: g * \tuple {s, t} = \tuple {s, t} }\)

Definition of Stabilizer

\(\ds \)

\(=\)

\(\ds \set {g \in G: \tuple {g *_S s, g *_T t} = \tuple {s, t} }\)

Definition of $*$

\(\ds \)

\(=\)

\(\ds \set {g \in G: g *_S s = s \land g *_T t = t}\)

Definition of Cartesian Product

\(\ds \)

\(=\)

\(\ds \set {g \in G: g \in \Stab s \land g \in \Stab t}\)

Definition of Stabilizer

\(\ds \)

\(=\)

\(\ds \Stab s \cap \Stab t\)

Definition of Set Intersection

$\blacksquare$

Sources

• 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 3.3$: Group actions and coset decompositions: Exercise $3$