Math 360, Fall 2014, Assignment 9

Mathematical proofs, like diamonds, are hard as well as clear, and will be touched with nothing but strict reasoning.

- John Locke, Second Reply to the Bishop of Worcester

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

1. Image (of a homomorphism).
2. Kernel (of a homomorphism).
3. Coset multiplication (for cosets of a normal subgroup).
4. Quotient group.
5. Canonical projection (from a group to its quotient).

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

1. Theorem concerning forward images and pre-images of subgroups (Thoerem 13.12).
2. Theorem characterizing monomorphisms in terms of kernels (Corollary 13.18).
3. Criteria for normality (Theorem 14.13).
4. Theorem concerning the normality of kernels (Corollary 13.20).

Solve the following problems:

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

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