Problems to submit for grading
Please organize your work, justify your steps, and write in complete sentences with correct punctuation. Once you finish, please follow the directions in the Canvas assignment to submit them. If you have any questions (about the math or writing or submission process or anything), please let me know!
Problem. D&F Section 3.3 #3.
Hint: Diamond Isomorphism Theorem.
Problem. Let $G$ be a group. Suppose that \(G\) has a normal subgroup \(N\) such that \(G/N\) is cyclic. Prove that for all subgroups \(H\le G\), if \(H\cap N= 1\), then \(H\) is cyclic.
Hint: Diamond Isomorphism Theorem.
Problem. Let $G$ be a group. Prove that \(G/Z(G)\) can not be isomorphic to the quaternion group \(Q_8\)1.
Hint: towards a contradiction, assume that \(G/Z(G) \cong Q_8\). Looking at the subgroup lattice for \(Q_8\), you see that it has two (in fact three) cyclic subgroups of order 4 that intersect in a subgroup of order 2. What does the Lattice Isomorphism Theorem allow you to deduce about \(G\)? Now keep pushing to a contradiction.
Problems for extra practice
Please try to solve some of these problems, but you do not turn them in. I’m happy to talk about these problems (or any others) if you have questions.
Problems. Work some problems from D&F Section 3.3. Here are some to consider: 8, 9
Footnotes
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For the definition of \(Q_8\), you can see D&F Section 1.5. The subgroup lattice of \(Q_8\) is given in Section 2.5 of D&F (on pg 69). ↩