Formal Definition
Let X be a non-empty set and G a group. Then X is called a G-space if it is equipped with an action of G on X. Note that automatically G acts by automorphisms (bijections) on the set. If X in addition belongs to some category, then the elements of G are assumed to act as automorphisms in the same category. Thus the maps on X effected by G are structure preserving. A homogeneous space is a G-space on which G acts transitively.
Succinctly, if X is an object of the category C, then the structure of a G-space is a homomorphism:
into the group of automorphisms of the object X in the category C. The pair (X,ρ) defines a homogeneous space provided ρ(G) is a transitive group of symmetries of the underlying set of X.
Read more about this topic: Homogeneous Space
Famous quotes containing the words formal and/or definition:
“On every formal visit a child ought to be of the party, by way of provision for discourse.”
—Jane Austen (1775–1817)
“... if, as women, we accept a philosophy of history that asserts that women are by definition assimilated into the male universal, that we can understand our past through a male lens—if we are unaware that women even have a history—we live our lives similarly unanchored, drifting in response to a veering wind of myth and bias.”
—Adrienne Rich (b. 1929)