Category:Group Actions

From ProofWiki
Jump to: navigation, search

This category contains results about Group Actions.
Definitions specific to this category can be found in Definitions/Group Actions.

Let $X$ be a set.

Let $\left({G, \circ}\right)$ be a group whose identity is $e$.

A (left) group action is an operation $\phi: G \times X \to X$ such that:

$\forall \left({g, x}\right) \in G \times X: g * x := \phi \left({\left({g, x}\right)}\right) \in X$

in such a way that the group action axioms are satisfied:

\((GA\,1)\)   $:$     \(\displaystyle \forall g, h \in G, x \in X:\) \(\displaystyle g * \left({h * x}\right) = \left({g \circ h}\right) * x \)             
\((GA\,2)\)   $:$     \(\displaystyle \forall x \in X:\) \(\displaystyle e * x = x \)             


This category has the following 4 subcategories, out of 4 total.





Pages in category "Group Actions"

The following 61 pages are in this category, out of 61 total.