One-parameter subgroup

Remarks

Despite the name, a one-parameter subgroup is the map $\gamma$, not its image.

Example: for a real $n \times n$ matrix $A$, the map $\gamma(t) = e^{tA}$ into $GL_n(\R)$, given by the matrix exponential, is continuous, and $e^{(s + t)A} = e^{sA} e^{tA}$ because $sA$ and $tA$ commute.

Non-example: $\gamma(t) = \begin{pmatrix} 1 & t \\ 0 & 1 + t^2 \end{pmatrix}$ is a continuous curve in $GL_2(\R)$ with $\gamma(0) = I$, but it is not a homomorphism. The $(2,2)$ entry of $\gamma(1)\gamma(1)$ is $4$, while that of $\gamma(2)$ is $5$.

If $G$ is a Lie group, every one-parameter subgroup is smooth. Then $\gamma \mapsto \gamma'(0)$ is a bijection from the one-parameter subgroups of $G$ onto the tangent space $T_eG$.