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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6320 • Algebra II

Fall 2025 • Question 2

Question

Math 6320 • Algebra II Fall 2025 • Question 2

Let \(\mathbb {Q}\) be the set of rational numbers, considered a group with respect to the usual addition. Prove that \(\mathbb {Q}\) is indecomposable, that is, it is not isomorphic to a direct sum \(A \oplus B\) of two non-trivial subgroups.

Raw LaTeX

Let \(\mathbb{Q}\) be the set of rational numbers, considered a group with
respect to the usual addition. Prove that \(\mathbb{Q}\) is indecomposable,
that is, it is not isomorphic to a direct sum \(A \oplus B\) of two non-trivial
subgroups.

Topic Tags

Finite Group Theory Normal Subgroups and Commutator Subgroups