Category:Definitions/Examples of Group Actions

This category contains definitions of examples of Group Action.

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:

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

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

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

A right group action is a mapping $\phi: X \times G \to X$ such that:

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

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

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

The group $G$ thus acts on the set $X$.

Subcategories

This category has the following 2 subcategories, out of 2 total.

The following 8 pages are in this category, out of 8 total.

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

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