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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Spring 2006 • Question 3

Question

Math 6310 • Algebra Spring 2006 • Question 3

Show that \[ \mathbb {Q}/\mathbb {Z}\cong \bigoplus _p(\mathbb {Q}/\mathbb {Z})_{(p)} \] where \(p\) runs over all primes numbers and \((\mathbb {Q}/\mathbb {Z})_{(p)}\) denotes the subgroup of elements of \(\mathbb {Q}/\mathbb {Z}\) whose order is a power of \(p\).

Raw LaTeX

Show that
\[
\mathbb{Q}/\mathbb{Z}\cong\bigoplus_p(\mathbb{Q}/\mathbb{Z})_{(p)}
\]
where \(p\) runs over all primes numbers and \((\mathbb{Q}/\mathbb{Z})_{(p)}\) denotes the subgroup
of elements of \(\mathbb{Q}/\mathbb{Z}\) whose order is a power of \(p\).

Topic Tags

Finite Group Theory Normal Subgroups and Commutator Subgroups