91Ó°ÊÓ

In \(S_{5}\) find an element of order \(n\), for all \(2 \leq n \leq 5\). Also determine the (cyclic) subgroup of \(S_{5}\) that each of these elements generates.

A pyramid has a square base and four faces that are equilateral triangles. If we can move the pyramid about (in three dimensions), how many nonequivalent ways are there to paint its five faces if we have paint of four different colors? How many if the color of the base must be different from the color(s) of the triangular faces?

a) Find all the elements of order 10 in \(\left(\mathrm{Z}_{40},+\right)\). b) Let \(G=\langle a\rangle\) be a cyclic group of order 40 . Which elements of \(G\) have order 10?

a) Find all generators of the cyclic groups \(\left(\mathbf{Z}_{12},+\right),\left(\mathbf{Z}_{16},+\right)\), and \(\left(\mathbf{Z}_{24},+\right)\). b) Let \(G=\langle a\rangle\) with \(o(a)=n\). Prove that \(a^{k}, k \in \mathbf{Z}^{+}\), generates \(G\) if and only if \(k\) and \(n\) are relatively prime. c) If \(G\) is a cyclic group of order \(n\), how many distinct generators does it have?