definition of group right action
Topics
About: group
About: set
The table of contents of this article
Starting Context
- The reader knows a definition of group.
- The reader knows a definition of set.
Target Context
- The reader will have a definition of group right action.
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.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\( G\): \(\in \{ \text{ the groups } \}\)
\( S\): \(\in \{ \text{ the sets } \}\)
\(*f\): \(: S \times G \to S\)
//
Conditions:
\(\forall g_1, g_2 \in G, \forall s \in S (f (f (s, g_1), g_2) = f (s, g_1 g_2))\)
\(\land\)
\(\forall s \in S (f (s, 1) = s)\)
//
\(f (s, g)\) is usually denoted as \(s g\).
2: Note
The notation, \(s g\), is permissible because \(s g_1 g_2\) is not ambiguous because of the conditions for group right action.
