DefinitionAdditionally, if we consider what is known as a group action, with G denoting a group and X a set, we have a description of the behaviour of the group G on a set X by assigning to each element of G a transformation of X. Formally speaking a group action is a function ⋅:G×X→X that sates a bunch of properties, like identity and compatibility. Group actions arise in algebra, geometry, and combinatorics to investigate symmetries, permutations, and properties of sets that are invariant under transformations by groups. Basis, Key to Understanding Algebraic Objects! Example : If G is a group, X is a set, a group action ⋅ satisfies e⋅x=x (where e is the identity element of G ) and (gh)⋅x=g⋅(h⋅x), for all g,h∈G, x∈X.
