Math 360, Fall 2014, Assignment 5

I was at the mathematical school, where the master taught his pupils after a method scarce imaginable to us in Europe. The proposition and demonstration were fairly written on a thin wafer, with ink composed of a cephalic tincture. This the student was to swallow upon a fasting stomach, and for three days following eat nothing but bread and water. As the wafer digested the tincture mounted to the brain, bearing the proposition along with it.

- Jonathan Swift, Gulliver's Travels

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

1. Subgroup generated by a subset.
2. Finitely generated group.
3. Permutation (of a set $A$).
4. Symmetric group on $A$ (we denoted this by $\mathrm{Sym}(A)$ in class; note that it is not the same thing as the "symmetry group of $A$," which we shall define later).

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

1. Theorem characterizing the order of $\mathrm{Sym}(A)$ when $A$ is a finite set.

Solve the following problems:

1. Section 8, problems 1, 4, 6, and 9.
2. Isometries of $\mathbb{R}^n$. Fix a positive integer $n$. The Euclidean distance function on $\mathbb{R}^n$ is the function $d:\mathbb{R}^n\times\mathbb{R}^n\rightarrow\mathbb{R}$ defined by the formula $$d((x_1,\dots,x_n),(y_1,\dots,y_n)) = \sqrt{(x_1-y_1)^2+\dots+(x_n-y_n)^2}.$$ A permutation $f$ of $\mathbb{R}^n$ is said to be an isometry if, for every pair of points $p,q\in\mathbb{R}^n$, we have $$d(f(p),f(q)) = d(p,q).$$ Give three examples of isometries, then show that the set of isometries is a subgroup of $\mathrm{Sym}(\mathbb{R}^n)$.

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Questions:

Is the theorem we were supposed to state encapsuled within Definition 8.6? A theorem number wasn't given and that was the closest that I could find.