752: Group Action

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definition of group action

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Topics

About: group
About: set

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Natural Language Description
• 3: Note

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Starting Context

• The reader knows a definition of group.
• The reader knows a definition of set.

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Target Context

• The reader will have a definition of group action.

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Orientation

There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

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Main Body

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1: Structured Description

Here is the rules of Structured Description.

Entities:
$G$: $\in \{\text{ the groups }\}$
$S$: $\in \{\text{ the sets }\}$
$\ast f$: $:G\times S\to S$
//

Conditions:
$\mathrm{\forall }{g}_1,g_2\in G,\mathrm{\forall }s\in S(f(g_2,f(g_1,s))=f(g_2{g}_1,s))$
$\wedge$
$\mathrm{\forall }s\in S(f(1,s)=s)$
//

$f(g,s)$ is usually denoted as $gs$.

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2: Natural Language Description

For any group, $G$, and any set, $S$, any map, $f:G\times S\to S$, such that $\mathrm{\forall }{g}_1,g_2\in G,\mathrm{\forall }s\in S(f(g_2,f(g_1,s))=f(g_2{g}_1,s))$ and $\mathrm{\forall }s\in S(f(1,s)=s)$, while $f(g,s)$ is usually denoted as $gs$

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3: Note

The notation, $gs$, is permissible because $g_2{g}_{1}s$ is not ambiguous because of the conditions for group action.

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References

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