# Problem Sheet E

This Problem Sheet is based on Lectures 8 and 9.

- Problem 1 asks you to prove that if $X$ is $ \varphi$-related to $Y$ then $ \theta_t^X \circ \varphi = \varphi \circ \theta_t^Y$.
- Problem 2 shows that the Lie bracket (or Lie derivative) measures the failure for two flows to commute.
- Problem 3 studies the
*symplectic linear group* - Problem 4 asks to show that an embedded Lie subgroup is automatically closed.
- Problem 5 asks you to show that an abelian Lie group has abelian Lie algebra (you will prove the converse on the next sheet).
- Problem 6 asks you identify the Lie bracket on $ \mathfrak{gl}(n)$.
- Problem 7 is a bit out of place… It will be useful in lectures to come.

Comments and questions?