Is the following example of one parameter group correct?

Question

In this book (An Introduction to Lie Groups and the Geometry of Homogeneous Spaces) the author gives the following example of one parameter group:

The map $\varphi(t) = e^t$ is a one-parameter subgroup of the additive Lie group $\Bbb R$.

Is this correct? because $$\varphi(t+s)=e^{t+s}=e^t.e^s\neq \varphi(t)+\varphi(s).$$

Answer 1

It's a mistake. The map $\varphi(t) = e^t$ is a one-parameter subgroup of the multiplicative group $\mathbb R^+$; and the map $\varphi(t)=t$ is a one-parameter subgroup of the additive group $\mathbb R$.