definition of effective group left action
Topics
About: group
The table of contents of this article
Starting Context
- The reader knows a definition of stabilizer of point with respect to group left action.
Target Context
- The reader will have a definition of effective group left 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:
\( G\): \(\in \{\text{ the groups }\}\)
\( S\): \(\in \{\text{ the sets }\}\)
\(*f\): \(: G \times S \to S\), \(\in \{\text{ the group left actions }\}\)
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Conditions:
\(\cap_{s \in S} G_s = \{1\}\)
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2: Note
Any free group left action is an effective group left action, because \(\forall s \in S (G_s = \{1\})\) implies \(\cap_{s \in S} G_s = \{1\}\), obviously, but not vice versa.
\(\cap_{s \in S} G_s = \{g \in G \vert \forall s \in S (g s = s)\}\), because for each \(g \in \cap_{s \in S} G_s\), for each \(s \in S\), \(g \in G_s\), so, \(g s = s\), so, \(g \in \{g \in G \vert \forall s \in S (g s = s)\}\); for each \(g \in \{g \in G \vert \forall s \in S (g s = s)\}\), for each \(s \in S\), \(g s = s\), so, \(g \in G_s\), so, \(g \in \cap_{s \in S} G_s\).
