<s>[INST] Find the order of the elements in the given groups I have to find the order of the following elements in the given groups:


*

*$(1 \ \ 2 \ \ 3) \ (1 \ \ 2\ \ 4) \text{ in } S_5$
$$\begin{pmatrix}
1 & 2  & 3 & 4 & 5\\ 
3 & 4 & 1 & 2 & 5
\end{pmatrix}=\begin{pmatrix}
1 & 3
\end{pmatrix}\begin{pmatrix}
2 & 4
\end{pmatrix}$$ order $=\operatorname{lcm}(2,2)=2$


Is it correct?


*

*$a^{45}$ in $G=\langle a\mid a^{140}=1\rangle$
$$\operatorname{ord}(a^{45})=\frac{140}{\gcd(45, 140)}=\frac{140}{5}=28$$


Is it right?


*

*$\begin{bmatrix}
0 & -1\\ 
1 & 0
\end{bmatrix} \text{ in } GL_2(\mathbb{R})$


$$A=\begin{bmatrix}
0 & -1\\ 
1 & 0
\end{bmatrix}  \\ A^2=\begin{bmatrix}
0 & -1\\ 
1 & 0
\end{bmatrix} \begin{bmatrix}
0 & -1\\ 
1 & 0
\end{bmatrix} =\begin{bmatrix}
-1 & 0\\ 
0 & -1
\end{bmatrix}  \\ A^3=\begin{bmatrix}
-1 & 0\\ 
0 & -1
\end{bmatrix} \begin{bmatrix}
0 & -1\\ 
1 & 0
\end{bmatrix} =\begin{bmatrix}
0 & 0\\ 
0 & 0
\end{bmatrix} $$
But how can I find the order?
 [/INST] Your first two questions are indeed correct.
Note that your calculation of $A^3$ is "off":
$$A^3= A^2 A=\begin{bmatrix}
-1 & 0\\ 
0 & -1
\end{bmatrix} \begin{bmatrix}
0 & -1\\ 
1 & 0
\end{bmatrix} =\begin{bmatrix}
0 & 1\\ 
-1 & 0
\end{bmatrix}$$
And $A^4 = A^2A^2 = -I\cdot -I = I$. Hence the order of $A = 4$.
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