Category:Group Actions

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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 $\struct {G, \circ}$ be a group whose identity is $e$.

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

$\forall \tuple {g, x} \in G \times X: g * x := \map \phi {\tuple {g, x} } \in X$

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

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


This category has the following 6 subcategories, out of 6 total.





Pages in category "Group Actions"

The following 35 pages are in this category, out of 35 total.