What is the order of $\langle A,B\rangle$? [closed]

I found out that

$\langle A ,B \rangle:= \langle A \cup B \rangle$

Suppose that $ord(\langle A \rangle)=n<\infty$ and $ord(\langle B \rangle)=m<\infty$. What is the order of $\langle A ,B \rangle$?

Solution 1:

The answer vary by the choice of $A$ and $B$. For example consider $\mathcal{D_4}$ symmetry group of a square. Take $A=r $ rotation by $\pi/2 $ and $B=r^2$. Then $ \text{ord}\left(A\right)=4 $ and $ \text{ord}\left(B\right)=2 $, and $ \text{ord}\left\langle A,B\right\rangle =|\left\{ id,r,r^{2},r^{3}\right\} |=4 $.

Now take $B=s$ where $s $ is a reflection with respect to the real axis, then also $ \text{ord}\left(B\right)=2$, but now $ \left\langle A,B\right\rangle =\mathcal{D}_{4} $ so that $ \text{ord}\left\langle A,B\right\rangle =8 $