I am looking at exercise 15.9 of Tu's "Introduction to manifolds" (2nd edition). Here is the text:
(a) For $r \in \mathbb{R}^\times := \mathbb{R} - \{0\}$, let $M_r$ be the $n \times n$ matrix $$M_r = \begin{bmatrix} r & & &\\ & 1 & &\\ & & ... & \\ & & & 1\end{bmatrix}=[re_1,e_2...e_n],$$ where $e_1,e_2...e_n$ is the standard basis for $\mathbb{R}^n$. Prove that the map $$f: GL(n,\mathbb{R})\rightarrow SL(n,\mathbb{R}) \times \mathbb{R}^{\times},$$ $$A \rightarrow (AM_{1/detA}, detA),$$ is a diffeomorphism.
(b) The center $Z(G)$ of a group $G$ is the subgroup of elements $g \in G$ that commute with all elements of $G$: $$Z(G):=\{g \in G| gx = xg\textrm{ for all }x \in G\}.$$ Show that the center of $GL(2, \mathbb{R})$ is isomorphic to $\mathbb{R}^\times$, corresponding to the subgroup of scalar matrices, and that the center of $SL(n,\mathbb{R}) \times \mathbb{R}^{\times}$ is isomorphic to $\{\pm1 \}\times\mathbb{R}^\times$. The group $\mathbb{R}^\times$ has two elements of order 2, while the group $\{\pm1 \}\times\mathbb{R}^\times$ has four elements of order 2. Since their centers are not isomorphic, $GL(n,\mathbb{R)}$ and $SL(n,\mathbb{R}) \times \mathbb{R}^{\times}$ are not isomorphic as groups.
(c) Show that $$h: GL(3,\mathbb{R}) \rightarrow SL(3, \mathbb{R}) \times \mathbb{R}^{\times},$$ $$A \rightarrow ((detA)^{(1/3)}A, det A),$$ is a Lie group isomorphism.
The same argument as in (b) and (c) prove that for $n$ even, the two Lie groups $GL(n,\mathbb{R)}$ and $SL(n,\mathbb{R}) \times \mathbb{R}^{\times}$ are not isomorphic as groups, while for $n$ odd, they are isomorphic as Lie groups.
Regarding (a) I believe that we need to show that $f$ is $C^\infty$ and has a $C^\infty$ inverse. Being the determinant smooth in $GL$ as well as matrix multiplication (indeed the group operation of $GL$ seen as a Lie group) I believe this is sufficient to prove that $A$ is smooth in $GL \times \mathbb{R}^{\times}$ (previously the smoothness of a Cartesian product of two smooth functions was proved). Being $SL$ a regular submanifold of $GL$ we can conclude that $f$ is smooth in $SL \times \mathbb{R}^{\times}$. As for the inverse, I believe we can define the function $$b: SL(n,\mathbb{R})\times \mathbb{R}^{\times} \rightarrow GL(n,\mathbb{R}),$$ $$(B, r) \rightarrow B M_r,$$ that to me maps back $A$ smoothly.
Regarding (b) I have proved by direct computation the fact that to belong to the center of $GL(2,\mathbb{R})$ a non singular matrix has to be a scalar one, i.e. written in the form $rI$, and, always directly, I have verified the group isomorphism $rI \rightarrow r$ that just follows from linearity. As for $SL(n,\mathbb{R})\times \mathbb{R}^{\times}$ I have observed that the only two non singular scalar matrices with $det = 1$ are $I$ and $-I$ (which correspond to $\pm1$) while $\mathbb{R}^{\times}$ is equal to its center. Same thing, the isomorphism can be proved directly. As for the official errata of the book, the sentence about the order of the elements should be "elements of order at most 2" in both occurrences. I see that the only elements with finite order (2 and 1 respectively) are $-1$ and $1$. However, in $SL(n,\mathbb{R})\times \mathbb{R}^{\times}$ we have $(-1, -1), (1, -1), (-1, 1), (1, 1)$, four elements of order 2. So the two centers (hence the two groups) cannot be isomorphic for the same reason. I was just wondering if I had to use (a) in any way.
Regarding (c) I believe that in general $(detA)^{(1/3)}A \notin SL(n,\mathbb{R})$ as $det((detA)^{(1/3)}A)=(detA)^2$ and maybe the correct formula is $$A \rightarrow ((detA)^{(-1/3)}A, det A).$$ With this assumption, I believe that we can find the inverse of h as $$h^{-1}: (S, r) \rightarrow (r^{1/3}S).$$ This proves the isomorphism (it can be verified directly), and it is clearly valid only for odd numbers as even square roots of negative real numbers are not defined in $\mathbb{R}^{\times}$.
Apart from the (possible) typos, which I would kindly ask to double check (did not find (c) in the official errata, but the problem with points b was already raised here) I was wondering what was the connection between (a) and (b, c). Either (a) is needed to prove (b) or (c), and I am missing something, or (a) was just introduced to show how there can be a diffeomorphism between the underlying manifolds of two Lie groups without implying that the groups itself are isomorphic. The diffeomorphism at (c) would not work as example as it is not valid for $n$ even.
What do you think? The question is long winded (apologies) but the doubts are quite coincise in essence. thanks