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.

- Can't say for sure, but I have a feeling it has something to do with "Cayley's Theorem." I don't really know why and don't totally understand it but, it's just a gut feeling I have. There were some key words in that proof like "order" and while the entire thing was talking about infinite groups, the last paragraph before the exercises mentions something about finite groups. Hope this helps.

Yes, I had in mind the observation that follows Definition 8.6: the order of $S_n$ is $n!$. (This follows Definition 8.6, but isn't part of it: the black rectangle marks the end of the definition.) - Steven.Jackson (talk) 14:43, 7 October 2014 (UTC)