Definitions for "Lie group"
An analytic manifold G that has a group structure such that that the map (x,y) - xy from G x G to G is analytic (i. e. infinitely differentiable). The general linear groups GL(), GL(), and their subgroups, are the most common examples. There is an extensive and deep theory of Lie groups, with many theoretical and applied uses.
A differentiable manifold that has a group structure on its elements, with the property that the group operations (multiplication and inversion) are continuous. A standard and useful example is the set of nonsingular real matrices. Other examples are the group of rigid motions in Euclidean space and the group of equivalence classes of transformations that agree on some neighborhood of the identity. The last was the original object of study by Sophus Lie around 1890.
a differentiable manifold obeying the group properties and that