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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.

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.