Wording of a group action

  • #1
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My textbook says the following: "Let ##G## be a group and ##G## act on itself by left conjugation, so each ##g \in G## maps ##G## to ##G## by ##x \mapsto gxg^{-1}##". I am confused by the wording of this. ##g## itself is not a function, so how does it map anything at all? I am assuming this is supposed to mean that ##g \cdot x = gxg^{-1}## is the definition of the action, but why does it use the map notation as if ##g## is a function, when ##g## is really an element?
 

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  • #2
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My textbook says the following: "Let ##G## be a group and ##G## act on itself by left conjugation, so each ##g \in G## maps ##G## to ##G## by ##x \mapsto gxg^{-1}##". I am confused by the wording of this. ##g## itself is not a function, so how does it map anything at all? I am assuming this is supposed to mean that ##g \cdot x = gxg^{-1}## is the definition of the action, but why does it use the map notation as if ##g## is a function, when ##g## is really an element?
It's ##\varphi : G \longrightarrow Inn(G)## with ##g \longmapsto \iota_g## where ##\iota_g : x \longmapsto gxg^{-1}## is the inner automorphism "conjugation by ##g##". ##\varphi## is a group homomorphism from ##G## to the group of inner automorphisms which is a subgroup of the automophism group ##Aut(G)## of ##G##. Whether you say "##G## acts on ##X##" or ##G \longmapsto Aut(X)## is a group homomorphism is the same thing. The notation "acts / operates on" with a dot is only a bit shorter than to introduce this homomorphism ##\varphi## and replace the dot by ##\varphi##.
 
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