I'm looking for an elegant way to identify and justify in writing the order of a group's element.
$ A \in (GL(2,\mathbb{R}), \; \cdot \;) \quad , \quad A= \left( \begin{array}{cc} 0&1\\ -1&1\\ \end{array} \right) $
$ord \; A := min\{ A^k = E_2 \; | \; k \in \mathbb{N} \backslash \{0\} \}$
I just started calculating $A$, $A^2$, $A^3$, ... and compared the result to $E_2$. So far so good, the result is $ord \; A = 6$.
But do I need to calculate and write down all prior powers to show that $A^k \neq E_2$ for all $k < 6$?
Lagrange doesn't help here either because $ord \; (GL(2,\mathbb{R}), \; \cdot \;) = \infty$ and therefore every $k \in \mathbb{N}\backslash \{0\}$ is divisor of $\infty$.