What type of matrices in $GL_2(\mathbb{C})$ have finite order?
In particular, I am trying to solve the following problem.
Find $A,B\in GL_2(\mathbb{C})$ such that $A$ and $B$ have finite order but $AB$ has an infinite order.
How to proceed?
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See here: https://math.stackexchange.com/questions/316030/finding-a-b-in-sl-2-bbbz-of-finite-order-with-the-property-that-ab-c-whe?rq=1 – Pedro Aug 12 '17 at 05:27
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1Possible duplicate of Finding $A,B\in SL_2(\Bbb{Z})$ of finite order with the property that $AB=C$ where the order of $C$ is infinite. – Glorfindel Aug 12 '17 at 08:43
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Any "reflection" matrix with eigenvalues $1$ and $-1$ will have finite order (viz. $2$). Such a matrix has trace zero and determinant $-1$ so has the form $$\pmatrix{a&b\\c&-a}$$ with $a^2+bc=1$. This gives a lot of possibilities for $A$ and $B$. Surely it can't be hard to write down some $A$ and $B$ such that $AB$ has infinite order?
Angina Seng
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