Skip to main content
\(\newcommand{\identity}{\mathrm{id}} \newcommand{\notdivide}{\nmid} \newcommand{\notsubset}{\not\subset} \newcommand{\lcm}{\operatorname{lcm}} \newcommand{\gf}{\operatorname{GF}} \newcommand{\inn}{\operatorname{Inn}} \newcommand{\aut}{\operatorname{Aut}} \newcommand{\Hom}{\operatorname{Hom}} \newcommand{\cis}{\operatorname{cis}} \newcommand{\chr}{\operatorname{char}} \newcommand{\Null}{\operatorname{Null}} \newcommand{\transpose}{\text{t}} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \setcounter{chapter}{-1}\)


  • Use the external direct product to construct new groups from old.
  • Find the orders of elements in a product of cyclic groups.
  • Determine when a product of cyclic groups is cyclic.
  • Classify up to isomorphism the multiplicative groups \(U(n)\text{.}\)


Subsection8.2.2Homework 8

Homework exercises are valuable preparation for your quiz on this material. Generally I do not collect homework for grading -- however, I will expect to review your homework progress in the event of your requesting a revision on a quiz. You are welcome to discuss homework problems with your groups and with the class, both in person and in the Slack #math channel.

  1. Exercises Section 8.4, #6.
  2. Exercises Section 8.4, #3.
  3. Exercises Section 8.4, #4.
  4. Up to isomorphism, list all groups of order 15 that can be built from direct products of cyclic groups.
  5. Up to isomorphism, list all groups of order 16 that can be built from direct products of cyclic groups.
  6. In the direct product group \(G = \mathbb{Z}_6 \oplus \mathbb{Z}_{12}\text{,}\) how many elements are there of order 24? (Try to avoid listing them all.)
  7. Let \(n\geq 2\text{.}\) Prove that there are at least two non-isomorphic groups of order \(n^2\text{.}\)
  8. Find a subgroup of \(G = \mathbb{Z}_{15} \oplus \mathbb{Z}_{12} \oplus \mathbb{Z}_4\) that has order 9. Is your subgroup cyclic?
  9. Express the multiplicative group \(G=U(165)\) as... (a) an external direct product of \(U\)-groups; and then as (b) an external direct product of cyclic groups of the form \(\mathbb{Z}_n\text{.}\)
  10. Give an example of an integer \(n\) for which \begin{equation*} U(n) \cong \mathbb{Z}_9 \oplus \mathbb{Z}_4 \oplus \mathbb{Z}_2. \end{equation*}
  11. Prove that \(U(140) \cong U(144)\text{.}\)
  12. Find a subgroup of the multiplicative group \(G=U(140)\) that's isomorphic to \(\mathbb{Z}_4\oplus \mathbb{Z}_6\text{.}\) Hint

    The trickiest part of this problem is being concrete about what the elements of this subgroup are -- not only what they are up to isomorphism with an appropriate \(\mathbb{Z}\)-group.

    Use the following fact about \(U\)-groups to accomplish this fact.

    For example, since \(120 = 40\cdot 3\) and 40 and 3 are relatively prime, we have

    \begin{equation*} U(120) \cong U(40) \oplus U(3) \end{equation*}

    and the \(U(40)\) factor is equal to

    \begin{equation*} U_3(120) = \{ 1, 4, 7, 10, 13, \ldots, 112, 115, 118\}. \end{equation*}

    (Shortcut: I built this set by beginning at 1, and then skip-counting by 3's.)

Subsection8.2.3Video Extra: Short Exact Sequences