91Ó°ÊÓ

Let \(\omega\) be the complex number \((1 / \sqrt{2})(1+i)\). a) Show that \(\omega^{8}=1\) but \(\omega^{n} \neq 1\) for \(n \in \mathbf{Z}^{+}, 1 \leq n \leq 7\). b) Verify that \(\left\\{\omega^{n} \mid n \in \mathbf{Z}^{+}, 1 \leq n \leq 8\right\\}\) is an abelian group under multiplication.

Let \(f: G \rightarrow H\) be a group homomorphism. If \(a \in G\) with \(\circ(a)=n\), and \(\sigma(f(a))=k\) (in \(H)\), prove that \(k \mid n\).

a) If \(H, K\) are subgroups of a group \(G\), prove that \(H \cap K\) is also a subgroup of \(G\). b) Give an example of a group \(G\) with subgroups \(H, K\) such that \(H \cup K\) is not a subgroup of \(G\)

a) Find all \(x\) in \(\left(\mathbf{Z}_{5}^{*}, \cdot\right)\) such that \(x=x^{-1}\). b) Find all \(x\) in \(\left(\mathbf{Z}_{11}^{*}, \cdot\right)\) such that \(x=x^{-1}\). c) Let \(p\) be a prime. Find all \(x\) in \(\left(\mathbf{Z}_{p}^{*}, \cdot\right)\) such that \(x=x^{-1}\). d) Prove that \((p-1) ! \equiv-1(\bmod p)\), for \(p\) a prime. [This result is known as Wilson's Theorem, although it was only conjectured by John Wilson (1741-1793). The first proof was given in 1770 by Joseph Louis Lagrange (1736-1813).]

a) Find \(x\) in \(\left(U_{g}, \cdot\right)\) where \(x \neq 1, x \neq 7\) but \(x=x^{-1}\). b) Find \(x\) in \(\left(U_{16}, \cdot\right)\) where \(x \neq 1, x \neq 15\) but \(x=x^{-1}\). c) Let \(k \in \mathbf{Z}^{+}, k \geq 3\). Find \(x\) in \(\left(U_{2^{k}}, \cdot\right)\) where \(x \neq 1\), \(x \neq 2^{k}-1\) but \(x=x^{-1}\)