One-parameter group

In mathematics, a one-parameter group or one-parameter subgroup usually means a continuous group homomorphism
from the real line formula_2 (as an additive group) to some other topological group formula_3. 
If formula_4 is injective then formula_5, the image, will be a subgroup of formula_3 that is isomorphic to formula_2 as an additive group.
One-parameter groups were introduced by Sophus Lie in 1893 to define infinitesimal transformations. According to Lie, an "infinitesimal transformation" is an infinitely small transformation of the one-parameter group that it generates. It is these infinitesimal transformations that generate a Lie algebra that is used to describe a Lie group of any dimension.
The action of a one-parameter group on a set is known as a flow. A smooth vector field on a manifold, at a point, induces a "local flow" - a one parameter group of local diffeomorphisms, sending points along integral curves of the vector field. The local flow of a vector field is used to define the Lie derivative of tensor fields along the vector field.
Examples.
Such one-parameter groups are of basic importance in the theory of Lie groups, for which every element of the associated Lie algebra defines such a homomorphism, the exponential map. In the case of matrix groups it is given by the matrix exponential.
Another important case is seen in functional analysis, with formula_3 being the group of unitary operators on a Hilbert space. See Stone's theorem on one-parameter unitary groups.
In his 1957 monograph "Lie Groups", P. M. Cohn gives the following theorem on page 58:
Physics.
In physics, one-parameter groups describe dynamical systems. Furthermore, whenever a system of physical laws admits a one-parameter group of differentiable symmetries, then there is a conserved quantity, by Noether's theorem.
In the study of spacetime the use of the unit hyperbola to calibrate spatio-temporal measurements has become common since Hermann Minkowski discussed it in 1908. The principle of relativity was reduced to arbitrariness of which diameter of the unit hyperbola was used to determine a world-line. Using the parametrization of the hyperbola with hyperbolic angle, the theory of special relativity provided a calculus of relative motion with the one-parameter group indexed by rapidity. The "rapidity" replaces the "velocity" in kinematics and dynamics of relativity theory. Since rapidity is unbounded, the one-parameter group it stands upon is non-compact. The rapidity concept was introduced by E.T. Whittaker in 1910, and named by Alfred Robb the next year. The rapidity parameter amounts to the length of a hyperbolic versor, a concept of the nineteenth century. Mathematical physicists James Cockle, William Kingdon Clifford, and Alexander Macfarlane had all employed in their writings an equivalent mapping of the Cartesian plane by operator formula_13, where formula_14 is the hyperbolic angle and formula_15.
In GL(n,ℂ).
An important example in the theory of Lie groups arises when formula_3 is taken to be formula_17, the group of invertible formula_18 matrices with complex entries. In that case, a basic result is the following:
It follows from this result that formula_4 is differentiable, even though this was not an assumption of the theorem. The matrix formula_21 can then be recovered from formula_4 as
This result can be used, for example, to show that any continuous homomorphism between matrix Lie groups is smooth.
Topology.
A technical complication is that formula_5 as a subspace of formula_3 may carry a topology that is coarser than that on formula_2; this may happen in cases where formula_4 is injective. Think for example of the case where formula_3 is a torus formula_33, and formula_4 is constructed by winding a straight line round formula_33 at an irrational slope.
In that case the induced topology may not be the standard one of the real line.