1.
Theorem: State the Criterion for a Product of Cyclic Groups to be Cyclic.
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Worksheet Weekly Practice 4
Instructions: You may type up or handwrite your work, but it must be neat, professional, and organized and it must be saved as a PDF file and uploaded to the appropriate Gradescope assignment. Use a scanner or scanning app to convert handwritten work on paper to PDF. I encourage you to type your work using the provided template.
All tasks below must have a complete solution that represents a good-faith attempt at being right to receive engagement credits. If your submission is complete and turned in on time, you will receive full engagement credit for the assignment. All other submissions will receive zero engagement credit. Read the guidelines at Grading Specifications carefully.
To abide by the class academic honesty policy, your work should represent your own understanding in your own words. If you work with other students, you must clearly indicate who you worked with in your submission. The same is true for using tools like generative AI although I strongly discourage you from using such tools since you need to build your own understanding here to do well on exams.
True/False, Multiple Choice, & Fill-In.
For these problems a justification is not required for credit, but it may be useful for your own understanding to include one. True/False problems should be marked True if the statement is always true, and False otherwise. Multiple choice problems may have more than one correct answer if that is indicated in the problem statement; be sure to select all that apply. Fill-in problems require a short answer such as a number, word, or phrase.
2.
3.
True/False: There are only two groups of order \(2p\) up to isomorphism.
Short Response.
Your responses to these questions should be complete solutions with justifications, as per the Grading Specifications.
4.
The dihedral group \(D_n\) of order \(2n\) for \(n\geq 3\) has a cyclic subgroup of \(n\) rotations and a subgroup of order \(2\text{.}\) Explain why \(D_n\) cannot be isomorphic to the external direct product of two such groups.
Problem Specs/Notes: This problem needs a clear description of a group property that is not shared between \(D_n\) and the product group that prevents the isomorphism for a Success.
5.
Find the largest order of a cyclic subgroup of \(\Z_6 \times \Z_{10}\times \Z_{15}\text{.}\) Give an example of such a subgroup. Is there a unique such subgroup?
Problem Specs/Notes: This problem needs a correctly derived order with examples provided for a Success.
6.
Construct the subgroup lattice for \(\Z_4\times \Z_2\text{,}\) justifying your work. It will be useful to use Lagrange’s Theorem, the Fundamental Theorem of Cyclic Groups, and the possible isomorphism classes of groups of size 2 and 4. Which subgroups are of the form \(H\times K \) where \(H\leq \Z_4 \) and \(K\leq \Z_2\text{?}\)
Problem Specs/Notes: This problem needs a correctly drawn lattice with clear justifications given for why it is complete for a Success.
7.
For each of the normal subgroups \(\langle (2,1)\rangle\) and \(\langle (0,1)\rangle\) of \(G=\Z_4\times \Z_2\text{,}\) carry out the following steps. Below I am writing \(N\) for the normal subgroup.
(a)
(b)
Create a Cayley table of the quotient group \(G/N\text{.}\)
(c)
Create a Cayley graph of \(G/N\text{.}\)
(d)
(e)
Consider the sublattice of the subgroup lattice of \(G\) consisting of all subgroups between \(G\) and \(N\text{.}\) What do you notice about the structure of this lattice?
Problem Specs/Notes: This problem needs all five responses for both normal subgroups for a Success.
8.
Let
\begin{equation*}
H=\left\{ \begin{bmatrix}
a \amp b \\ 0 \amp d
\end{bmatrix} \mid a,b,d\in \R, ad\neq 0\right\}\text{.}
\end{equation*}
Is \(H\) a normal subgroup of \(\GL(2,\R)\text{?}\)
Problem Specs/Notes: This problem needs careful computation of the matrix multiplications for a Success.
