Math 360, Fall 2016, Assignment 7

Mathematicians are like Frenchmen: whatever you say to them they translate into their own language, and forthwith it is something entirely different.

- Goethe

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

1. Permutation (of a set $S$).
2. Two-row notation (for permutations).

Solve the following problems:

1. Section 8, problems 1, 3, 5, 7, and 9 (Hint: for problems 7 and 9, start by drawing the cycle diagrams for the permutations in question.)
2. Draw the subgroup diagrams for $\mathbb{Z}_{20}$ and $\mathbb{Z}_{60}$.
3. 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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