I'm confused about how to go about generating the cyclic subgroups of given matricies.
In a question I'm doing I have to find what matrices does the intersection of 2 cyclic subgroups of matrices contain both unsure how to do so.
Q) Which matrices does H ∩ K contain? What is |H ∩ K|? When K denotes the (cyclic) subgroup of $GL(2, \Bbb C$) generated by X and when H denotes the (cyclic) subgroup generated by Y.
X=$\bigl( \begin{smallmatrix} \theta & 0 \\ 0 & \theta^{-1} \end{smallmatrix} \bigr)$ and Y= $\bigl( \begin{smallmatrix} 0 & -1 \\ 1 & 0 \end{smallmatrix} \bigr)$ where $\theta = e^{\frac {πi}{3}}$
Is there a method for finding the cyclic subgroup? I found this Is this group of matrices cyclic? but I'm not sure if it applies here. I think I can find the intersection if I could just know how to get the matrices
Best Answer
A cyclic group (in particular, a subgroup of some other group) is a group generated by some element (in our case, matrix) $A$. This means that such group includes elements $e$ (the identity element, in our case the identity matrix), $A$, $A^2$, $A^3$ and so on until we get $e$ again. So write down powers of $X$ and $Y$ until you get the identity matrix, and you'll get all the matrices from $K$ and $H$ respectively.