One-parameter group

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In mathematics, a one-parameter group or one-parameter subgroup usually means a continuous group homomorphism

${\displaystyle \varphi :\mathbb {R} \rightarrow G}$

from the real line ${\displaystyle \mathbb {R} }$ (as an additive group) to some other topological group ${\displaystyle G}$. If ${\displaystyle \varphi }$ is injective then ${\displaystyle \varphi (\mathbb {R} )}$, the image, will be a subgroup of ${\displaystyle G}$ that is isomorphic to ${\displaystyle \mathbb {R} }$ 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.^[1] 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.