Problem Set 1

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Worksheet Problem Set 1

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.

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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.

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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.

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Problems Doable After Week 1.

1. Left-Right Cancellation Implies Commutativity.

Let \(G\) be a group with the property that for any \(x,y,z \in G\text{,}\) \(xy=zx\) implies \(y=z\text{.}\) Prove that \(G\) is Abelian.

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Problem Specs/Notes: This problem needs careful choice of group elements to apply the assumption to for a Success.

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2. One-Step Subgroup Test.

Prove the One-Step Subgroup Test: Let \(G\) be a group and \(H\) a nonempty subset of \(G\text{.}\) If \(ab^{-1}\) is in \(H\) whenever \(a\) and \(b\) are in \(H\text{,}\) then \(H\leq G\text{.}\)

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You may use the Two-Step Subgroup Test in your proof.

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3. Playing with Group Elements.

Choose one of the problems below to complete for this Problem Set.

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(a)

If \(a\) and \(b\) are group elements and \(ab\neq ba\text{,}\) prove that \(aba\neq e\text{.}\)

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(b)

If \(a\) and \(b\) are distinct group elements, prove that either \(a^{2}\neq b^{2}\) or \(a^{3}\neq b^{3}\text{.}\)

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Problem Specs/Notes: This problem needs careful choice of proof method for a Success.

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Problems Doable After Week 2.

4. Order is Preserved Under Conjugation.

Let \(G\) be a group and \(a,b\in G\text{.}\) Prove that \(|aba^{-1}|=|b|\text{.}\)

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Problem Specs/Notes: For a Success you need to handle:

The finite order case, which may require a two-sided proof.
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The infinite order case.
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5. Cyclic Subgroups.

Prove that \(H=\left\{ \begin{bmatrix}1 \amp n \\ 0 \amp 1\end{bmatrix}\mid n \in \Z \right\}\) is a cyclic subgroup of \(\GL(2,\R)\text{.}\)

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Problem Specs/Notes: This problem needs justification of both the fact that the given set is a subgroup and that it is cyclic for a Success. Depending on your argument structure, it may be possible to do both at once.

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6. Well-Foundedness of Even and Odd Permutations.

Prove using induction that a permutation in \(S_n\) cannot be expressed both as the product of both an even number of transpositions and an odd number of transpositions. In other words, that our definitions of even and odd permutations are well-founded.

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Hint.

It might also be helpful to first reduce the argument to the case of the identity permutation.

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Problem Specs/Notes: This problem must be done with induction, giving clear base case and inductive step, for a Success.

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