Math 360, Fall 2013, Assignment 8

Do not imagine that mathematics is hard and crabbed and repulsive to commmon sense. It is merely the etherealization of common sense.

- Lord Kelvin

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

1. Direct product (of two groups).
2. Direct sum (of two abelian groups written additively).
3. Isometry (of $\mathbb{R}^n$).
4. Symmetry group (of a subset of $\mathbb{R}^n$).

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

1. Theorem relating $\mathbb{Z}_m\times\mathbb{Z}_n$ to $\mathbb{Z}_{mn}$ (Theorem 11.5 in the text).
2. Fundamental Theorem of Finitely Generated Abelian Groups (Theorem 11.12).

Solve the following problems:

1. Section 11, problems 1, 3, 8, 23, and 25.
2. Section 12, problems 1, 5, and 7.

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