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\( \newcommand{\cat}[1] {\mathrm{#1}} \newcommand{\catobj}[1] {\operatorname{Obj}(\mathrm{#1})} \newcommand{\cathom}[1] {\operatorname{Hom}_{\cat{#1}}} \newcommand{\multiBetaReduction}[0] {\twoheadrightarrow_{\beta}} \newcommand{\betaReduction}[0] {\rightarrow_{\beta}} \newcommand{\betaEq}[0] {=_{\beta}} \newcommand{\string}[1] {\texttt{"}\mathtt{#1}\texttt{"}} \newcommand{\symbolq}[1] {\texttt{`}\mathtt{#1}\texttt{'}} \newcommand{\groupMul}[1] { \cdot_{\small{#1}}} \newcommand{\groupAdd}[1] { +_{\small{#1}}} \newcommand{\inv}[1] {#1^{-1} } \newcommand{\bm}[1] { \boldsymbol{#1} } \require{physics} \require{ams} \require{mathtools} \)
Math and science::Algebra::Aluffi

Groups. Group action.

Consider a group and a target object, such as a set. A group actions associate each group element with [what?] of a target object, such as a set.

Group action

Let \( G \) be a group with operation \( m_G : G \times G \to G \), and let \( A \) and object. A function \( \sigma : G \to \operatorname{Auto}(A) \) is a group action iff:

  1. the identity of \( G \) maps to the identity of \( A \).
  2. either:
    1. for any \( g, h \in G \), \( \sigma(\; m_G(g, h)\;) \; = \; \) [\( \;\; ? \; \circ \; ? \;\; \) ] [left-action]
    2. for any \( g, h \in G \), \( \sigma(\; m_G(g, h)\;) \; = \; \) [\( \;\; ? \; \circ \; ? \;\; \) ] [right-action]

Why are there two types of actions?