Left Coset of Stabilizer in Group of Transformations

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Let $S$ be a non-empty set.

Let $G$ be a group of permutations of $S$.

Let $t \in G$.

Let $G_t$ be the set defined as:

$G_t = \set {g \in G: \map g t = t}$

Then each left coset of $G_t$ in $G$ consists of the elements of $G$ that map $t$ to some element of $S$.


Let $x \in G$.

Consider the left coset $x G_t$.

Let $\map x t = s$.


\(\ds y\) \(\in\) \(\ds x G_t\)
\(\ds \leadstoandfrom \ \ \) \(\ds y^{-1} x\) \(\in\) \(\ds G_t\) Element in Left Coset iff Product with Inverse in Subgroup
\(\ds \leadstoandfrom \ \ \) \(\ds \map {y^{-1} } {\map x t}\) \(=\) \(\ds t\) Definition of $G_t$
\(\ds \leadstoandfrom \ \ \) \(\ds \map {y^{-1} } s\) \(=\) \(\ds t\) Definition of $s$
\(\ds \leadstoandfrom \ \ \) \(\ds \map y t\) \(=\) \(\ds s\)