Dummit and Foote define a group action as follows:

A group action of a group G on a set A is a map from G x A to A (written as g a, for all g G and a A) satisfying the following properties

(1) ( a) = ( ) a for all , G and all a A

(2) 1 a = a for all a A

Dummit and Foote Section 1.7 Group Actions Exercise 14 reads as follows:

Let G be a group and let A = G. SHow that if G is non-abelian then the maps defined by g a = ag for all g, a G dosatisy the axioms of a (left) group action of G on itselfnot

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I am unsure of whether the operation of the group G on the set G - that is g a can be treated the same as the operation of two elements of G as a group - that is ga.

Can anyone clarify this? [I suspect it cannot!]

Anyway, proceeding withthe problem we need to show

(1) ( a) = ( ) a for all , G and all a A is NOT satisfied

Then ( a) = (a )

The term on the right above is strictly a group action operation of on a set element a

How do I progress this?

I do not think I can write ( a) as there is a group action mixed with a group operation ???

Am I overthinking this?

Peter