A one-parameter subgroup of a Lie group 
is a smooth group homomorphism 
. Thus 
and 
is the identity element.
Every element 
of the Lie algebra 
determines exactly one such group
homomorphism with 
, given by 
using the exponential
map (Etingof 2020, pp. 44-45).
A one-parameter subgroup need not be injective. For the unit circle as a Lie
group, 
has period 
.
The term refers to this parametrized group homomorphism,
whose image is a subgroup
of 
.
