Stabilizer of Cartesian Product of Group Actions

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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:

\(\displaystyle \Stab {s, t}\) \(=\) \(\displaystyle \set {g \in G: g * \tuple {s, t} = \tuple {s, t} }\) Definition of Stabilizer
\(\displaystyle \) \(=\) \(\displaystyle \set {g \in G: \tuple {g *_S s, g *_T t} = \tuple {s, t} }\) Definition of $*$
\(\displaystyle \) \(=\) \(\displaystyle \set {g \in G: g *_S s = s \land g *_T t = t}\) Definition of Cartesian Product
\(\displaystyle \) \(=\) \(\displaystyle \set {g \in G: g \in \Stab s \land g \in \Stab t}\) Definition of Stabilizer
\(\displaystyle \) \(=\) \(\displaystyle \Stab s \cap \Stab t\) Definition of Set Intersection

$\blacksquare$


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