A _one-parameter group_ or a _one-parameter subgroup_ is as a [map](Maps.md) to a [[Lie groups]], $\mbox{A}:\mathbb{R} \rightarrow \mbox{GL}(n;\mathbb{C}),$ which we refer to as a _one-parameter subgroup of [$\mbox{GL}(n;\mathbb{C})$](GL(n;F).md)_. If $\mbox{A}(.)$ is a one parameter subgroup of $\mbox{GL}(n;\mathbb{C})$, then there exists a unique complex matrix $X$ such that the elements of the one-parameter group are [matrix exponential](Matrix%20exponentials.md)s, $A(t) = e^{tX}\;\; \forall t,s \in \mathbb{R}$. %%Is this a linear group? I suspect that it's linear%% # Properties of one-parameter groups For a [map](Maps.md) to be a one parameter subgroup, the following properties must be met: 1) $A(t)$ is [continuous](Continuity.md) 2) $A(0) = \mathbb{1}$ 3) $A(t+s) = A(t)A(s) \;\; \forall t,s \in \mathbb{R}$ %%This mention of continuity is an issue. This should preferably link to a note or a subnote to continuous group homomorphisms%% # Lie Algebra An element $X$ is in a [Lie algebras](Lie%20algebras.md) $\mathfrak{g}$ if and only if the [One-parameter groups](One-parameter%20groups.md) that may be constructed with $X$ is entirely in the corresponding [[Lie groups]], $\mbox{G}.$ # Examples * [One-parameter unitary groups](One-parameter%20unitary%20groups.md)s #MathematicalFoundations/Algebra/AbstractAlgebra/GroupTheory/Lie/LieGroups/Algebras/LieAlgebras #MathematicalFoundations/Algebra/AbstractAlgebra/GroupTheory/Lie/LieGroups