Stabilizers are Subgroups

Stabilizers are Subgroups

$G_{s} \leq G$ (see Stabilizers).

Proof

\begin{aligned} \Rightarrow g^{- 1} \cdot (g \cdot x) & = g^{- 1} \cdot x \\ \Rightarrow (g^{- 1} g) \cdot x & = g^{- 1} \cdot x \\ \Rightarrow x & = g^{- 1} \cdot x \end{aligned}

So then $g^{- 1} \in G_{x}$ .
3. (products) Suppose $g_{1}, g_{2} \in G_{x}$ . Then:

\begin{aligned} (g_{1} g_{2}) \cdot x & = g_{1} \cdot (g_{2} \cdot x) \\ = g_{1} \cdot x \\ = x \end{aligned}

So $g_{1}, g_{2} \in G_{x}$ .

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