Category:Definitions/Group Actions

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This category contains definitions related to Group Actions.
Related results can be found in Category:Group Actions.


Let $X$ be a set.

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


Left Group Action

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:

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


Right Group Action

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 3 subcategories, out of 3 total.

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D

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Pages in category "Definitions/Group Actions"

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