# Left Coset of Stabilizer in Group of Transformations

Jump to navigation
Jump to search

## Theorem

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

- 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): Chapter $7$: Cosets and Lagrange's Theorem: Exercise $15$