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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Spring 2007 • Question 8

Question

Math 6310 • Algebra Spring 2007 • Question 8

Let \(N\) be a positive integer. Consider the natural map from \(\operatorname {SL}(2,\mathbb {Z})\) to \(\operatorname {SL}(2,\mathbb {Z}/N)\) obtained by reducing each matrix entry modulo \(N\). Prove or disprove: this map is surjective.

Raw LaTeX

Let \(N\) be a positive integer. Consider the natural map from \(\operatorname{SL}(2,\mathbb{Z})\)
to \(\operatorname{SL}(2,\mathbb{Z}/N)\) obtained by reducing each matrix entry modulo \(N\).
Prove or disprove: this map is surjective.

Topic Tags

Matrix Similarity, Jordan Forms, and Finite-Order Diagonalization