1. Let $\Phi (n)=\{ i | 0\leq i \leq n-1 s.t. gcd(n,i)=1 \}$ and
let $\phi (n) =| \Phi (n) |$. Thus $\phi (n)$ is the number of
numbers between $0$ and $n-1$ which are co-prime with $n$. $\phi $
is also called the Euler-phi function.
(i) Show that if $gcd(m,n)=1$, then $\phi (mn) =\phi (m) \phi
(n)$.
(ii) Get an expression for $\phi (n)$, when $n=p_1^{n_1}
\ldots p_k^{n_k }$.
(iii) Show that $n=\sum_{k divides n} \phi (k) $.
(iv) Let $Z_n $ act on the $n$-gon. Compute the cycle polynomial
of this action.
2. Compute the cycle polynomial of the group of symmetry of the
cube acting on its faces.
3. You are given two square cardboard pieces and square vacant
slots. In how many ways can you put the cardboard pieces into the
slots? Generalize?
4. Let $\sigma $ be a permutation on the set $\{ 1,2, \ldots
,n\}$. We construct the matrix $P_{\sigma }$ such that $P_{\sigma
}(i,j)=1 $ iff $\sigma (i) =j$, and zero otherwise. Show that the
collection of all such matrices, where $\sigma $ ranges over all
permutations, is a subgroup of $GL_n $. What the determinant
values for such matrices? What matrices have determinant $1$?
5. How many permutations are there on $n$ letters with $n_1$
cycles of length $1$, $n_2 $ cycles of length $2$ and so on?
Verify, for $n=3,4,5$.