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.

--------------------End of assignment--------------------

Solutions:

Definitions:

1. Direct Product of Two Groups.

   Definition:

   Take two groups \(G\) and \(H\). Take their Cartesian product \(G\times H\). Define the binary operation \(\cdot\) on this set as:$$ (a,b)\cdot(c,d) = (ac,bd) $$

   Where \(a,c\in G\) and \(b,d\in H\). The binary structure \(G\times H, \cdot\) is the direct product of \(G\) and \(H\).

   Example:

   The direct product of \(\mathbb{Z}_2\) and \(]mathbb{Z}_3\) is \((0,0),(0,1), (0,2), (1,0), (1,1), (1,2)\)

   Non-Example:

   \(\mathbb{Z}_4\) is not the direct product of \(\mathbb{Z}_2\) and \(\mathbb{Z}_2\).