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:

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

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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 $\mathbb{Z}_2$.
Direct Sum of Two Abelian Groups
Definition:
A direct sum is a just a direct product of abelian groups - groups written additively.
Example:
$\mathbb{Z}\times\mathbb{Z}$ is the direct sum of $mathbb{Z}$.
Non-Example:
$GL(n,\mathbb{R})\times GL(n,\mathbb{R})$ is not a direct sum - the underlying group is not abelian.
Isometry of $\mathbb{R}^n$
Definition:
An isometry of $\mathbb{R}^n$ is a permutation (bijective function) of $mathbb{R}^n$ that preserves distance. If $\phi$ is an isometry, then:$$ d(\phi(x),\phi(y)) = d(x,y)$$
Example:
The translation $f:\mathbb{R}^3\rightarrow \mathbb{R}^3$ given by:$$ f(x,y,z) = (x+1,y+1,z+1) $$

is a isometry.

Non-Example:

The function \(f(x,y,z) = (x^3,y,z)

is not an isometry.