definition of group right action that corresponds to group left action
Topics
About: group
About: set
The table of contents of this article
Starting Context
- The reader knows a definition of group left action.
- The reader knows a definition of group right action.
Target Context
- The reader will have a definition of group right action that corresponds to 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:
\( G\): \(\in \{ \text{ the groups } \}\)
\( S\): \(\in \{ \text{ the sets } \}\)
\( f\): \(: G \times S \to S\), \(\in \{\text{ the group left actions }\}\)
\(*f'\): \(: S \times G \to S, (s, g) \mapsto f (g^{-1}, s)\), \(\in \{\text{ the group right actions }\}\)
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Conditions:
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2: Note
Let us see that \(f'\) is indeed a group right action.
For each \(g_1, g_2 \in G\) and each \(s \in S\), \(f' (f' (s, g_1), g_2) = f' (f (g_1^{-1}, s), g_2) = f (g_2^{-1}, f (g_1^{-1}, s)) = f (g_2^{-1} g_1^{-1}, s) = f ((g_1 g_2)^{-1}, s) = f' (s, g_1 g_2)\).
For each \(s \in S\), \(f' (s, 1) = f (1^{-1}, s) = f (1, s) = s\).
Note that \(f\) itself cannot be regarded to be any group right action; we have constructed the group right action from \(f\): 'group left action' is not about just denoting \(G \times S\) instead of \(S \times G\): \(f'': S \times G \to S, (s, g) \mapsto f (g, s)\) is not any group right action, because \(f'' (f'' (s, g_1), g_2) = f'' (f (g_1, s), g_2) = f (g_2, f (g_1, s)) = f (g_2 g_1, s) = f'' (s, g_2 g_1)\), which does not equal \(f'' (s, g_1 g_2)\) in general.
