91Ó°ÊÓ

A root of unity in the complex numbers is a number \(\omega\) such that \(\omega^{n}=1\) for some positive integer \(n .\) We then say that \(\omega\) is an \(n\) -th root of unity. Describe thé set of \(n\) -th roots of unity in \(\mathbf{C}\). Show that this set is a cyclie group of order \(n .\)

Let \(f: G \rightarrow G^{\prime}\) be a homomorphism with kernel \(H .\) Assume that \(G\) is finite. Show that order of \(G=\) (order of image of \(f\) )(order of \(H\) ). Compare with the analogous theorem on dimensions of linear maps.

Let \(\mathbf{C}^{x}\) be the multiplicative group of complex numbers. What is the kernel of the homomorphism absolute value $$ \text { of } \mathbf{C}^{x} \text { into } \mathbf{R}^{\times} ? $$

Let \(H\) be a subgroup of \(G\), and assume that \(x H x^{-1}=H\) for all \(x \in G\). Then \(x^{-1} H x=H\) for all \(x \in G\), and \(H x=x H\) for all \(x \in G\).

Let \(G\) be a group and \(H\) a subgroup. Show that \(H\) is normal if and only if \(x H x^{-1}=H\) for all \(x \in G\).

Let \(G\) be a group, and \(a, b, c\) be elements of \(G .\) If \(a b=a c\), show that \(b=c\).

Consider the additive group of integers \(\mathbf{Z}\). Show that it has only two generators, namely 1 and \(-1 .\) In general, show that an infinite cyclic group has only two generators.

Let \(G\) be a finite abelian group of order \(n\), and let \(a_{1}, \ldots, a_{\mathrm{n}}\) be its elements. Show that the product \(a_{1} \cdots a_{n}\) is an element whose square is the unit element.

Let \(S_{3}\) be the symmetric group, and let \(\epsilon: S_{3} \rightarrow\\{1,-1\\}\) be the homomorphism given by the sign of the permutation. What is the order of the kernel of \(\epsilon ?\)

Show that a finite group whose order is a prime number is necessarily cyclic.