Math 360, Fall 2013, Assignment 9

The danger already exists that the mathematicians have made a covenant with the devil to darken the spirit and to confine man in the bonds of Hell.

- Saint Augustine

Carefully define the following terms, then give one example and one non-example of each:

1. Homomorphism.
2. Monomorphism.
3. Epimorphism.
4. Trivial homomorphism.
5. Projection homomorphism (Example 13.8 in the text).
6. Reduction modulo $n$.
7. Image (of a set under a function).
8. Pre-image (of a set under a function).
9. Fiber (of a homomorphism over a point).
10. Kernel (of a homomorphism).
11. Factor group (of a group $G$ by a normal subgroup $H$).
12. Automorphism.
13. Inner automorphism.
14. Conjugation (by an element of a group).

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

1. Theorem concerning images and pre-images of subgroups (Theorem 13.12).
2. Characterization of fibers as cosets (Theorem 13.15).
3. Characterization of monomorphisms (Corollary 13.18).
4. Fundamental theorem on homomorphisms (Theorem 14.11).
5. Criteria for normality (Theorem 14.13).

Solve the following problems:

1. Section 13, problems 1, 3, 7, 8, 17, 47, and 48.
2. Section 14, problems 1, 5, 11, 23, and 24.

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