Math 360, Fall 2016, Assignment 13

"Reeling and Writhing, of course, to begin with," the Mock Turtle replied; "And then the different branches of Arithmetic - Ambition, Distraction, Uglification, and Derision."

- Lewis Carroll, Alice's Adventures in Wonderland

Carefully state the following theorems (you do not need to prove them):

1. Theorem concerning the normality of kernels.
2. Fundamental Theorem of Homomorphisms.
3. Theorem relating monomorphisms to subgroups.
4. Theorem relating epimorphisms to quotient groups.
5. Theorem relating the image, kernel, and domain of a general homomorphism (a.k.a. "generalized rank-nullity theorem").
6. Chinese Remainder Theorem.

Solve the following problems:

1. Section 11, problems 1, 2, 3, and 16 (hint for 16: first ask yourself whether $\mathbb{Z}_{12}$ is isomorphic to $\mathbb{Z}_3\times\mathbb{Z}_4$).
2. Section 13, problems 25, 26, 27, 44, and 45.
3. Section 14, problems 3, 11, and 24.

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