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# Group Action on Itself: Multiply Left/Right, Adjoint/Conjugation

ABSALG-$UFS5W Let$G$be a group with neutral element$e$and composition law$\ast$. Let$G^{\rm \,op}$, called the opposite to$(G, \ast)$, be the group whose set of elements equals the set$G$, but whose group composition law$\bar{\ast}$is$h\bar{\ast}g:= g\ast h$. If$G$is abelian, then$(G,\ast)=(G^{\rm \,op},\bar{\ast})$, so we assume in this question that$G$is not necessarily abelian. Let${\rm Aut}(G)$be the group of automorphisms of$G$, with composition law$\circ$given by composition of maps. An action of a group$(G, \ast)$on a set$X$is a map$g\mapsto \alpha_g$from$G$to the set of bijective maps from$X$to$X$, such that$\alpha_e(x)=x$, for all$x\in X$, and$\alpha_{h\ast g}=\alpha_h\circ\alpha_g$, all$h,g\in G$, where$\circ$is composition of maps. For$g\in G$, let$L_g:G\rightarrow G$be the map$L_g(h)=g\ast h$, let$R_g:G\rightarrow G$be the map$R_g(h)=h\ast g$, and let${\rm Ad}_g$be the map${\rm Ad}_g(h)=g\ast h\ast g^{-1}$, for all$h\in G$. We call the$L_g$left multiplications, the$R_g$right multiplications, and the${\rm Ad}_g$adjoint maps, a.k.a. conjugations. 1) The map between sets$g\mapsto {\rm Ad}_g$, for$g\in G$, . 2) The map between sets$g\mapsto {\rm Ad}_{g^{-1}}$, for$g\in G$, . 3) The map$g\mapsto L_g$, for$g\in G$, . 4) The map$g\mapsto R_g$, for$g\in G\$,

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