Left Coset of Stabilizer in Group of Transformations

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Theorem

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 $S$.


This article, or a section of it, needs explaining.
In particular: The source work does not discuss group actions, but still defines $G_t$ as the stabilizer of $t$ in $G$. This needs to be reviewed and put into the language of group actions as a result related to transformation group action -- but this area of group theory is not as well covered in $\mathsf{Pr} \infty \mathsf{fWiki}$ as it ought to be. I need to dig out my college notes on group actions, which were more comprehensive and understandable than any of the other works I have on my shelf, which will also need to be exploited properly.
Hence the second part of this question in Whitelaw is not covered yet.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{Explain}} from the code.


Proof

Let $x \in G$.

Consider the left coset $x G_t$.

Let $\map x t = s$.

Then:

\(\ds y\) \(\in\) \(\ds x G_t\)
\(\ds \leadstoandfrom \ \ \) \(\ds y^{-1} x\) \(\in\) \(\ds G_t\) 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 $s$
\(\ds \leadstoandfrom \ \ \) \(\ds \map y t\) \(=\) \(\ds s\)

$\blacksquare$


Sources

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