Math 360, Fall 2017, Assignment 5

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

Read:

1. Section 5.

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

1. Group of units (of a monoid).
2. $\mathrm{Sym}(S)$ (for some set $S$).
3. Subgroup.
4. Subgroup generated by a subset.

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

1. Theorem concerning intersections and unions of subgroups.

Solve the following problems:

1. Section 5, problems 1, 2, 3, 4, 5, 6, 7, 8, 11, 12, 22, and 23.
2. 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 $\phi(\sigma)$ really is an element of $\mathrm{Sym}(B)$, then show that $\phi$ is an isomorphism.)

--------------------End of assignment--------------------