homework 21 (due May 4)

due date

Tuesday, May 4

instructions

Read Chapter 9 of our textbook up to and including Example 5. Complete the following exercises:

1ac, 3, 18, 19

some hints

1a. Since $[S_4:A_4]=2$, what can you say about the normality of $A_4$?

1b. Show that $D_4$ is not normal by finding a left coset that is not equal to a right coset.

3. See your notes for the subgroup lattice of $Q_8$. Do some brute-force work to see which subgroups are actually normal. Don't forget that if the index is 2, you get normality for free. Theorem 8.3 should be useful to classify the factor groups up to isomorphism.

18. We did an example in class that will answer this question for you. Look for a non-cyclic group $G$ that has a normal cyclic subgroup $N$ such that $G/N$ is cyclic.

19. See the back of the book.

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