Quotients of Lie Groups
If G is a Lie group and N is a normal Lie subgroup of G, the quotient G / N is also a Lie group. In this case, the original group G has the structure of a fiber bundle (specifically, a principal N-bundle), with base space G / N and fiber N.
For a non-normal Lie subgroup N, the space G / N of left cosets is not a group, but simply a differentiable manifold on which G acts. The result is known as a homogeneous space.
Read more about this topic: Quotient Group
Famous quotes containing the words lie and/or groups:
“Mind not the old man beseeching the young man;
Let not the child’s voice be heard, nor the mother’s entreaties;
Make even the trestles to shake the dead, where they lie awaiting
the hearses,
So strong you thump, O terrible drums—so loud you bugles blow.”
—Walt Whitman (1819–1892)
“Belonging to a group can provide the child with a variety of resources that an individual friendship often cannot—a sense of collective participation, experience with organizational roles, and group support in the enterprise of growing up. Groups also pose for the child some of the most acute problems of social life—of inclusion and exclusion, conformity and independence.”
—Zick Rubin (20th century)