Finding elements in GL_2(R) with specific orders

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SUMMARY

The discussion centers on finding elements in the group GL2(ℝ) with a specific order, particularly an element A such that o(A) = 165. The participant Jack inquires about systematic methods to achieve this, rather than relying on trial and error. A suggested approach involves using a rotation matrix corresponding to an angle of 2π/165, which may yield the desired order through matrix multiplication.

PREREQUISITES
  • Understanding of group theory, specifically matrix groups.
  • Familiarity with GL2(ℝ) and its properties.
  • Knowledge of matrix multiplication and its implications on order.
  • Basic concepts of rotations in two-dimensional space.
NEXT STEPS
  • Research methods for determining the order of matrices in GL2(ℝ).
  • Explore the construction of rotation matrices and their properties.
  • Study the implications of matrix products on the order of resulting matrices.
  • Investigate other systematic approaches for finding elements of specific orders in matrix groups.
USEFUL FOR

Mathematicians, students of abstract algebra, and anyone interested in group theory and matrix analysis will benefit from this discussion.

jackmell
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Hi,

Is there a way to find elements in ##GL_2(\mathbb{R})## with an arbitrary order other than by trial and error? Suppose I wanted to find ##A\in GL_2(\mathbb{R}):\; o(A)=165##. Is there no methodical way to find this? Is it possible perhaps I can find a product of matricies, each individually relatively easy to determine the order, such that the order of the product is the order I need?

Thanks,
Jack
 
Physics news on Phys.org
How about a rotation by an angle of [itex]2\pi/165[/itex]?
 
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