Suppose I have Givens matrices (as defined here), $G_1,...G_k$, and some permutation $P$ which mapping $\{1,..., k\}$ to $\{1,..., k\}$. Is it true that:
$$G_1G_2... G_k = G_{p(1)}G_{p(2)}...G_{p(k)}$$
The reason I think this true is that I heard they represent rotations and rotations can be applied in any order. If this is not true then what is the flaw in my logic here?
Doubt should be eliminated through experimentation. Here we examine two Given's rotations given by $$G_{13} = \begin{bmatrix} 0 & 0 & -1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{bmatrix}, \quad G_{23} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0 \end{bmatrix}. $$ Let $x = \begin{bmatrix} x_1 & x_2 & x_3 \end{bmatrix}^T$. It is clear that $$G_{13} x = \begin{bmatrix} -x_3 \\ x_2 \\ x_1 \end{bmatrix}, \quad G_{23} G_{13} x = \begin{bmatrix} -x_3 \\ -x_1 \\ x_2 \end{bmatrix},$$ whereas $$G_{23} x = \begin{bmatrix} x_1\\ -x_3 \\ x_2 \end{bmatrix}, \quad G_{13} G_{23} x = \begin{bmatrix} -x_2 \\ -x_3 \\ x_1 \end{bmatrix}.$$ We conclude that $G_{23}G_{13}x = G_{13} G_{23} x$ if and only $x_1 = x_2 = x_3$ or equivalently $x_j = \lambda \in \mathbb{R}$ for $j=1,2,3$. It follows that Given's rotations do not commute in general.
Rotations around a common axis commute. In the example given above $G_{13}$ rotates around the $x_2$ axis, whereas $G_{23}$ rotates around the $x_1$ axis.