definition of group action
Topics
About: group
About: set
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
- 3: Note
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 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:
\( G\): \(\in \{ \text{ the groups } \}\)
\( S\): \(\in \{ \text{ the sets } \}\)
\(*f\): \(: G \times S \to S\)
//
Conditions:
\(\forall g_1, g_2 \in G, \forall s \in S (f (g_2, f (g_1, s)) = f (g_2 g_1, s))\)
\(\land\)
\(\forall s \in S (f (1, s) = s)\)
//
\(f (g, s)\) is usually denoted as \(g s\).
2: Natural Language Description
For any group, \(G\), and any set, \(S\), any map, \(f: G \times S \to S\), such that \(\forall g_1, g_2 \in G, \forall s \in S (f (g_2, f (g_1, s)) = f (g_2 g_1, s))\) and \(\forall s \in S (f (1, s) = s)\), while \(f (g, s)\) is usually denoted as \(g s\)
3: Note
The notation, \(g s\), is permissible because \(g_2 g_1 s\) is not ambiguous because of the conditions for group action.
