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