homework 23 (due May 11)

due date

Tuesday, May 11

instructions

Read Section 9.3 of our textbook. Complete the following textbook exercise:

32

Also, complete the following exercises not in our textbook:

1. Prove each of the following:

(a) $Q_8/\langle -1\rangle\cong V_4$

(b) $D_4/\langle r^2 \rangle\cong V_4$

(c) $\mathbb{Z}/n\mathbb{Z}\cong \mathbb{Z}_n$

2. Let $G$ and $H$ be two groups and consider the direct product $G\times H$. Prove that there exists a subgroup $K$ of $G\times H$ such that $(G\times H)/K\cong G$.

some hints

Textbook exercise

32. For the "forward" direction of the iff proof, you need to show that $ker(\phi)$ only contains the identity. If $\phi$ is 1-1, how many elements can map to the identity in $H$? For the "reverse" direction, you need to assume that $ker(\phi)=\{e\}$ and show that $\phi$ is 1-1. There are several ways to accomplish this. One approach follows the standard attack for proving that a function is 1-1, and another can approach the problem using the First Isomorphism Theorem.

Non-Textbook exercises

1. For (a), (b), and (d), you should construct a homomorphism from the "numerator" of the quotient group to the set of the right that has the "denominator" of the quotient group as the kernel. Then apply the First Isomorphism Theorem. See your notes from Tuesday, May 4 for a homomorphism from $Q_8$ to $V_4$ that has kernel equal to $\langle -1\rangle$.

2. Mimic the very last example that I did class on Thursday.

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