Does a Group Action Always Use the Group's Original Operation?

[Bachelier]

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.

 

[Stephen Tashi]

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.

 

[xaveir]

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].

 

[Bachelier]

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.

 

[blue_raver22]

The action of a group ##G## on a set ##X## is not performed under the original operation of ##G##, but rather under a new operation defined by the map ##\phi##. This new operation is called the group action and it allows us to study the symmetries of the set ##X## using the group ##G##. This concept is fundamental in many areas of mathematics and has applications in various fields such as physics, chemistry, and computer science.

The fact that different sources may define the action using different operations does not change the fundamental definition of a group action. As long as the two operations satisfy the conditions listed above, they are both valid ways of defining the group action. However, it is important to be consistent within a specific context and use a single operation when studying a particular group action.

In summary, the group action is a new operation defined by the map ##\phi## and is not performed under the original operation of the group ##G##. It is a powerful tool for understanding the symmetries of a set and has many applications in various fields.

 

[blue_raver22]

The action of a group ##G## on a set ##X## is performed using the original operation of the group ##G##. This means that the map ##\phi## uses the group operation to combine elements of ##G## with elements of ##X##. This is consistent with the definition of a group action, which requires the map ##\phi## to preserve the group operation.

It is possible that the book you were reading uses a different operation to define the group action, but this would not be a standard definition. In order for a group to act on a set, the group operation must be used. Using a different operation would not satisfy the conditions for a group action as stated in the question.

It is important to note that the group action does not change the group operation itself. It is simply a way for the group to act on a set, and the group operation remains the same. This allows for the study of symmetries and transformations of a set using the principles of group theory.

In conclusion, the group action on a set ##X## is performed using the original operation of the group ##G##, as stated in the definition. This is a fundamental concept in group theory and is consistent with the definition of a group action.