Math 360, Fall 2015, Assignment 6

I tell them that if they will occupy themselves with the study of mathematics, they will find in it the best remedy against the lusts of the flesh.

- Thomas Mann, The Magic Mountain

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

1. Greatest common divisor (of two integers).
2. Permutation (of a set $A$).
3. Symmetric group (on a set $A$).
4. $S_n$.

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

1. Theorem concerning subgroups of cyclic groups.
2. Theorem characterizing when $\left\langle a\right\rangle = \left\langle b\right\rangle$ in $\mathbb{Z}$.
3. Theorem characterizing when $\left\langle a\right\rangle = \left\langle b\right\rangle$ in $\mathbb{Z}_n$.
4. Theorem concerning the order of $S_n$.
5. Theorem characterizing the values of $n$ for which $S_n$ is non-abelian.

Solve the following problems:

1. Section 6, problems 5, 23, 25, 33, and 34.
2. Section 7, problems 1, 3, and 5.
3. Section 8, problems 1, 3, 4, 5, 7, and 9.
4. Suppose $A$ and $B$ are sets with the same cardinality. Show that $\mathrm{Sym}(A)$ is isomorphic to $\mathrm{Sym}(B)$. (Hint: suppose $f:A\rightarrow B$ is a bijection. Define a map $\phi:\mathrm{Sym}(A)\rightarrow\mathrm{Sym}(B)$ by the formula $(\phi(\sigma))(b) = f(\sigma(f^{-1}(b)))$. Show that $\phi(\sigma)$ is actually a permutation of $B$, then show that $\phi$ is an isomorphism.)

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