Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Spring 2007 • Question 1

Question

Math 6310 • Algebra Spring 2007 • Question 1

Let \(F\) be a field. Let \(M_5(F)\) denote the set of 5-by-5 matrices with entries in \(F\), and let \(\mathcal {N}\) denote the subset of nilpotent matrices, \[ \mathcal {N}:=\{A\in M_5(F)\mid \text {there exists }N\text { such that }A^N=0\}. \] Let \(X\) denote the subset of \(M_5(F)\) consisting of those matrices which are squares of nilpotent matrices, \[ X:=\{M\in M_5(F)\mid \text {there exists }A\in \mathcal {N}\text { such that }M=A^2\}. \] Observe that \(G=\operatorname {GL}(5,F)\) acts on \(X\) by conjugation. Determine, with proof, the number of orbits for the action of \(G\) on \(X\).

Let \(F\) be a field. Let \(M_5(F)\) denote the set of 5-by-5 matrices with entries in \(F\), and let
\(\mathcal{N}\) denote the subset of nilpotent matrices,
\[
\mathcal{N}:=\{A\in M_5(F)\mid \text{there exists }N\text{ such that }A^N=0\}.
\]
Let \(X\) denote the subset of \(M_5(F)\) consisting of those matrices which are squares of
nilpotent matrices,
\[
X:=\{M\in M_5(F)\mid \text{there exists }A\in\mathcal{N}\text{ such that }M=A^2\}.
\]
Observe that \(G=\operatorname{GL}(5,F)\) acts on \(X\) by conjugation. Determine, with proof,
the number of orbits for the action of \(G\) on \(X\).

Topic Tags

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