I am reading the book: Lie Groups, Lie Algebras, and Representations: An Elementary Introduction by Brian C. Hall. I am stuck at the following exercise: exercise 11, chapter 2 . Can you help me?
Suppose $G$ is a matrix lie group in $GL(n, \mathbb{C})$ with the lie algebra $\mathfrak{g}$. suppose that $A$ is in G and that $||A-I||<1$,so the power series for $\log A$ is convergent. Is it necessarily the case that $\log A$ is in $\mathfrak{g}$? prove or give a counterexample.
In this book, $\log A$ is defined as a power series: $$ log A = \sum _{n=1}^{\infty} (-1)^{n+1}(A-I)^n/n$$ whenever the series converges.
I know that if $A$ is small enough, then we can prove that it is true.