One-parameter group

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a one-parameter group or one-parameter subgroup usually means a <a href="/facts/Continuous_(topology)/sKbl02pB">continuous</a> <a href="/facts/Group_homomorphism/sDguaNYs">group homomorphism</a>

φ
        :
        
          R
        
        →
        G
      
    
    {\displaystyle \varphi :\mathbb {R} \rightarrow G}

from the <a href="/facts/Real_line/6eq42xX5">real line</a> 
 
 
 
 
 R
 
 
 
 {\displaystyle \mathbb {R} }
 
 (as an <a href="/facts/Abelian_group/FfWwmx4R">additive group</a>) to some other <a href="/facts/Topological_group/cAgNMMRU">topological group</a> 
 
 
 
 G
 
 
 {\displaystyle G}
 
. 
If 
 
 
 
 φ
 
 
 {\displaystyle \varphi }
 
 is <a href="/facts/Injective/5ES6tqf5">injective</a> then 
 
 
 
 φ
 (
 
 R
 
 )
 
 
 {\displaystyle \varphi (\mathbb {R} )}
 
, the image, will be a subgroup of 
 
 
 
 G
 
 
 {\displaystyle G}
 
 that is isomorphic to 
 
 
 
 
 R
 
 
 
 {\displaystyle \mathbb {R} }
 
 as an additive group.
One-parameter groups were introduced by <a href="/facts/Sophus_Lie/CplQVa8S">Sophus Lie</a> in 1893 to define <a href="/facts/Infinitesimal_transformation/XCN1WpGS">infinitesimal transformations</a>. 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 <a href="/facts/Lie_algebra/n2gAUkNq">Lie algebra</a> that is used to describe a <a href="/facts/Lie_group/a9jCQyjF">Lie group</a> of any dimension.
The <a href="/facts/Action_(group_theory)/Vh9VtUUw">action</a> of a one-parameter group on a set is known as a <a href="/facts/Flow_(mathematics)/CrmJPfeV">flow</a>. A smooth vector field on a manifold, at a point, induces a local flow - a one parameter group of local diffeomorphisms, sending points along <a href="/facts/Integral_curve/wGbVcymt">integral curves</a> of the vector field. The local flow of a vector field is used to define the <a href="/facts/Lie_derivative/1KSJvrzk">Lie derivative</a> of tensor fields along the vector field.

One-parameter group open-in-new

One-parameter group