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Section Daily Prep 11
You will learn the definition of a group action and practice showing that a particular operation is or is not a group action.
Objectives: Basic Learning Objectives
Before our class meeting, you should use the resources below to be able to learn the following. You should be reasonably fluent with these; we’ll answer some questions on them in class but not reteach them in detail.
State and instantiate the definition of a (left) group action
Given a function
\(G\times X \to X\text{,}\) determine if it is an action of
\(G\) on
\(X\text{.}\)
Subsection Resources for Learning
Use these resources to prepare for class and answer the questions below.
Figure 115. Reference Video for Group Actions
Subsection Important Terms
Definition 116 . Group Action.
Let \(X\) be a set and \(G\) be a group. A (left) group action of \(G\) on \(X\) is a map \(G\times X \to X\text{,}\) \((g,x)\mapsto g\cdot x\) such that
\(e\cdot x= x\) for all
\(x\in X\)
\((gh)\cdot x=g\cdot(h\cdot x)\) for all
\(x\in X\) and
\(g,h\in G\)
We call \(X\) a G-set when it is equipped with an action of \(G\text{.}\)
