Stabilizer is Normal iff Stabilizer of Each Element of Orbit

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Theorem

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

Let $S$ be a set.

Let $*: G \times S \to S$ be a group action.

Let $x \in S$.

Let $\Stab x$ denote the stabilizer of $x$ under $*$.

Let $\Orb x$ denote the orbit of $x$ under $*$.


Then $\Stab x$ is normal in $G$ if and only if $\Stab x$ is also the stabilizer of every element in $\Orb x$.


Proof


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Necessary Condition

Let $\Stab x$ be normal in $G$.

Let $t \in \Stab x$.

Let $y \in \Orb x$.

By Definition 1 of Orbit (Group Theory):

$(1): \quad y = g * x$

for a $g \in G$.

By Definition 1 of Normal Subgroup:

${\Stab x} \circ g = g \circ \Stab x$

In particular:

$(2): \quad t \circ g = g \circ t'$

for a $t' \in \Stab x$.

Therefore:

\(\ds t * y\) \(=\) \(\ds t * \paren {g * x}\) $(1)$
\(\ds \) \(=\) \(\ds \paren {t \circ g} * x\) Group Action Axiom $\text {GA} 1$
\(\ds \) \(=\) \(\ds \paren {g \circ t'} * x\) $(2)$
\(\ds \) \(=\) \(\ds g * \paren {t' * x}\) Group Action Axiom $\text {GA} 1$
\(\ds \) \(=\) \(\ds g * x\) as $t' \in \Stab x$
\(\ds \) \(=\) \(\ds y\) $(1)$

That is:

$t \in \Stab y$

$\Box$


Sufficient Condition

Let $\Stab x$ be the stabilizer of every element in $\Orb x$.

We shall show Definition 3 of Normal Subgroup, i.e.

$\forall g \in G : g^{-1} \circ \Stab x \circ g \subseteq \Stab x$
$\forall g \in G : g \circ \Stab x \circ g^{-1} \subseteq \Stab x$

It suffices to verify the first inclusion, from which the second one follows by choosing $g^{-1}$ instead of $g$.


To this end, let $g \in G$.

Let $t \in \Stab x$.

By Definition 1 of Orbit (Group Theory):

$g * x \in \Orb x$

Thus by hypothesis:

$(3): \quad t \in \Stab {g * x}$

Therefore:

\(\ds \paren {g^{-1} \circ t \circ g} * x\) \(=\) \(\ds g^{-1} * \paren {t * \paren {g * x} }\) Group Action Axiom $\text {GA} 1$
\(\ds \) \(=\) \(\ds g^{-1} * \paren {g * x}\) by $(3)$
\(\ds \) \(=\) \(\ds \paren {g^{-1} \circ g} * x\) Group Action Axiom $\text {GA} 1$
\(\ds \) \(=\) \(\ds e * x\) Definition of Inverse Element
\(\ds \) \(=\) \(\ds x\) Group Action Axiom $\text {GA} 2$

That is:

$g^{-1} \circ t \circ g \in \Stab x$

$\blacksquare$


Sources

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