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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Fall 1996 • Question 8

Question

Math 6310 • Algebra Fall 1996 • Question 8

Let \(f\colon \mathbb {Z}^n\to \mathbb {Z}^n\) be a map of free Abelian groups of rank \(n\) defined by a matrix \(A\) with non-zero determinant. Show that the cokernel of \(f\) is finite and that its order is equal to the absolute value of the determinant of \(A\).

Raw LaTeX

Let \(f\colon \mathbb{Z}^n\to\mathbb{Z}^n\) be a map of free Abelian groups of rank \(n\) defined by
a matrix \(A\) with non-zero determinant. Show that the cokernel of \(f\) is finite and that its
order is equal to the absolute value of the determinant of \(A\).

Topic Tags

Abelian Groups and Classification of Finite Abelian Groups Finite Group Theory Free Groups and Group Presentations Matrix Similarity, Jordan Forms, and Finite-Order Diagonalization