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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Fall 2006 • Question 1

Question

Math 6310 • Algebra Fall 2006 • Question 1

Let \(F\) be a field. Let \(X\) denote the set of 8-by-8 matrices \(A\) with entries in \(F\) such that \(A^3=0\) but \(A^2\ne 0\). Observe that \(G=\operatorname {GL}(8,F)\) acts on \(X\) by conjugation. Compute the number of orbits for the action of \(G\) on \(X\).

Raw LaTeX

Let \(F\) be a field. Let \(X\) denote the set of 8-by-8 matrices \(A\) with entries in \(F\) such that
\(A^3=0\) but \(A^2\ne 0\). Observe that \(G=\operatorname{GL}(8,F)\) acts on \(X\) by conjugation.
Compute the number of orbits for the action of \(G\) on \(X\).

Topic Tags

Finite Group Theory Matrix Similarity, Jordan Forms, and Finite-Order Diagonalization