# Questions (doubts) on: Group Action on Manifolds

There are 2 questions that are bugging me in differential topology and I'd be glad if the same could be cleared up:

1. Let $X = x\frac{\partial}{\partial y}$ be a vector field on $M = R^2$, where $R$ is the set of reals. Find $W$ and $\theta : W \rightarrow M$ defining a $local$ 1-parameter subgroup whose infinitesimal generator is $X$.

2. Let $M = GL(2, R)$ ($R$ is set of real no.'s). Define a 1-parameter action on $M$ by

$\theta(t, A) = $$\begin{bmatrix} 1 & t \\ 0 & 1 \end{bmatrix}$$A$.

Here $A$ belongs to $GL(2,R)$ and the multiplication is the usual matrix multiplication.

2a) Show that $\theta$ is a 1-parameter group action.

2b) Find the infinitesimal generator.