Operator of the group action

  • Thread starter Bachelier
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  • #1
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Main Question or Discussion Point

A group ##G## is said to act on a set ##X## when there is a map ##\phi:G×X \rightarrow X## such that the following conditions hold for any element ##x \in X##.

1. ##\phi(e,x)=x## where ##e## is the identity element of ##G##.

2. ##\phi(g,\phi(h,x))=\phi(gh,x) \ \ \forall g,h \in G##.

My question is: is this action on the set ##X## performed under the operation of the group ##G## or under a different new operation. Only the ##Wikipedia## article author defines this operation as the group ##G## original operation. On the other hand, I was reading a different book and it defines the action using a totally new operation. Mind you this book is quite old.
 

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  • #2
Stephen Tashi
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My question is: is this action on the set ##X## performed under the operation of the group ##G## or under a different new operation.
Are you asking whether the notation [itex] gh [/itex] refers to the multiplication operation of the group [itex] G [/itex]? It does.
 
  • #3
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Dummit and Foote clearly define it using the operation of that same group. What book are you using?

Are you suggesting that your book defines a group action of [itex](G,\ast_1)[/itex] on [itex]X[/itex] via a function [itex]\phi : G \times X \to X[/itex] such that [itex]\phi(g,\phi(h,x))=\phi(g\ast_2h,x)[/itex], where [itex]g,h\in G; x\in X,[/itex] and where [itex]\ast_2[/itex] is the operation of another group defined on the elements of G? Because this definitely be a typo, as this would simply correspond to the usual definition of a group action of [itex](G,\ast_2)[/itex] on [itex]X[/itex].
 
  • #4
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I tend to agree with you guys. It seems the operation is not really that important though as long as the action on the set is well-defined.

For the sake of discussion, I am including an image of the Author's (A.J Green) definition.
 

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