How to find the order of a matrix?

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SUMMARY

The discussion focuses on determining the order of the Heisenberg group over a field, specifically examining a 3x3 upper triangular matrix H(F) with ones on the diagonal. The user concludes that raising this matrix to any power does not yield the identity matrix, indicating that the order of H(F) is infinite. The concept of infinite order is clarified, stating that if g raised to any positive integer m does not equal 1, then g possesses infinite order.

PREREQUISITES
  • Understanding of matrix theory, particularly upper triangular matrices.
  • Familiarity with group theory concepts, specifically the definition of order in groups.
  • Knowledge of the Heisenberg group and its properties.
  • Basic linear algebra skills, including matrix multiplication and identity matrices.
NEXT STEPS
  • Study the properties of the Heisenberg group in detail.
  • Learn about matrix exponentiation and its implications in group theory.
  • Explore examples of finite and infinite order elements in various groups.
  • Investigate the relationship between matrix representations and group orders.
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Mathematicians, students of abstract algebra, and anyone studying group theory and matrix representations will benefit from this discussion.

xsw001
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For example the Heisenberg group over Field.

H(F) is a 3x3 upper right triangle where the entries on the main diagonals are all 1's.

So by definition that I need to use this matrix and raise to the power where it becomes the identity matrix, then the number of that power would the order of H(F). But if I take the upper right triangle with all 1's in the main diagonal, it would never becomes identity matrix though no matter how many times I multiple them.

I couldn't find any examples from the material that I have. Am I doing something wrong here? Any suggestions?
 
Physics news on Phys.org
If you have g^m\ne1 for every positive integer m, you say that g has infinite order.
 

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