2.3 Group Theory 19 Direct product Let GandG′ be two groups. The direct product of GandG′, denoted by G×G′, is the group with elements (g,g′) ∈G×G′ for g ∈Gand g′ ∈G′, and the binary operation◦ : G×G′ →G×G′ such that (g,g′)◦(h,h′)= (g ◦h,g′ ◦h′). Lie group ALie group is a group where Gis a smooth manifold. This means that it can be described in a local scale with a set of smooth parameters and that one can interpolate smoothly between elements of G. Moreover, the group operation and its inverse are smooth. Group action Let Abe a set and Ga group. A group action of Gon Ais a functionGA : G×A→Athat has the properties 1. GA(e,a)=a for all a ∈A, 2. GA(g, (GA(g′,a))=GA(g ◦g′,a) for all g,g′ ∈Ganda ∈A. To avoid notational clutter, we often write GA(g,a)=g · a where the set Aon whichg ∈Gacts can be inferred from context. The group action properties can then be written as 1. e· a =a for all a ∈A, 2. g · (g′ · a)=(g ◦g′) · a for all g,g′ ∈Ganda ∈A. Free action The action of GonAis calledfree if no non-trivial element of G fixes a point of A, i.e. if g · a =a for some a ∈Aimplies that g =e, the identity element of G. Transitive action The action of GonAis calledtransitive if for every pair of elements a,a′ ∈Athere exists a g ∈Gsuch that g · a =a′. Regular action The action of Gon Ais called regular if it is both transitive and free, i.e. if for every pair of elements a,a′ ∈Athere exists a unique g ∈G such that g · a =a′.
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