1.
Prove that if \(K\) is a subgroup of \(G\) and \(\phi:G\to \bar{G}\) is an isomorphism, then \(\phi(K)=\{\phi(k)\mid k \in K\}\) is a subgroup of \(\bar{G}\text{.}\)
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Worksheet Problem Set 2
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 credits 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.
Problems Doable After Week 3.
Problem Specs/Notes: This problem needs careful choice of elements to apply the isomorphism properties to for a Success.
2.
Prove that a group that has more than one subgroup of order 5 must have order at least 25.
Problems Doable After Week 4.
3.
Prove or disprove that \(\Z \times \Z\) is a cyclic group.
Problem Specs/Notes: This problem needs either an identification of a generator or a clear example of why no generator exists for a Success.
4.
Let \(p\) be prime. Find a subgroup \(H\) of \(\Z_{p^2}\times \Z_{p^2}\) such that \((\Z_{p^2}\times \Z_{p^2})/H\) is isomorphic to \(\Z_p\times \Z_p\text{.}\)
Problem Specs/Notes: This problem could be approached either by constructing the isomorphism directly, in which case you must show that it is well-defined as a map on quotients, or by using the First Isomorphism Theorem, if you attempt it after we cover that result. In either case, it must be clear what subgroup \(H\) is.
5.
Choose one of the two statements below to prove.
(a)
Prove that a quotient group of a cyclic group is cyclic.
(b)
Prove that a quotient group of an Abelian group is Abelian.
Problem Specs/Notes: Don’t over-complicate this one!
6.
Let \(\phi:G \to \overline{G}\) be an isomorphism. Prove that if \(H \normaleq G\) then \(\phi(H)\normaleq \overline{G}\text{.}\)
Problem Specs/Notes: This problem needs careful choice of elements to apply the isomorphism properties to for a Success.
Problems Doable After Week 5.
7.
Chose one of the theorems below to prove.
(a)
(Second Isomorphism Theorem) If \(K\) is a subgroup of \(G\) and \(N\) is a normal subgroup of \(G\text{,}\) prove that \(K/(K\cap N) \cong KN/N\text{.}\) This is sometimes called the Diamond Isomorphism Theorem because it says that the upper left and lower right quotients of the diagram below are isomorphic.
Problem Specs/Notes: You do not need to prove that \(K\cap N\) is a subgroup of \(G\) or that \(KN\) is a subgroup when \(N\) is normal. You should argue either that \(N\) is normal in \(KN\) or \(K\cap N\) is normal in \(K\) and apply the First Isomorphism Theorem with an appropriately chosen homomorphism.
(b)
(Third Isomorphism Theorem) If \(M\) and \(N\) are normal subgroups of \(G\) with \(N\leq M\text{,}\) prove that \((G/N)/(M/N)\cong G/M\text{.}\)
Problem Specs/Notes: Note: by \(M/N\) here we mean the set
\begin{equation*}
\{mN \in G/N \mid m \in M\}\text{.}
\end{equation*}
You should apply the First Isomorphism Theorem with an appropriately chosen homomorphism.
Note: These are important results and their proofs are a good reminder of how we prove isomorphism results in algebra, so I would recommend attempting the one you don’t turn in for credit as well.
8.
Let \(G\) be an Abelian group of order 16. Suppose that there are elements \(a\) and \(b\) in \(G\) with \(|a|=|b|=4\) and \(a^2\neq b^2\text{.}\) Determine the isomorphism class of \(G\text{.}\)
Problem Specs/Notes: This problem needs a careful analysis of the possible Abelian groups of order 16 for a Success. Solutions that just give an argument that shows that the correct isomorphism class has the given properties without explaining why the other classes do not will not be considered successful.
9.
Let \(\phi: G \to \overline{G}\) be a surjective group homomorphism. Let \(H\) be a normal subgroup of \(G\) and suppose \(\phi(H)=\overline{H}\text{.}\) Prove or disprove that \(G/H \cong \overline{G}/\overline{H}\text{.}\)
Problem Specs/Notes: This problem needs careful thinking about the details for a Success - there are some subtleties.
