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Magazine Problem 5
Every complex number \(a+b \mathbf{i}\) may be represented as a point in the complex plane. The unit circle in the complex plane consists of all the complex numbers whose distance from the origin is 1 ; thus, clearly, the unit circle consists of all the complex numbers which can be written in the form $$ \cos x+\mathbf{i} \sin x $$ for some real number \(x\). Use the FHT to conclude that \(T \cong \mathbb{R} /\langle 2 \pi\rangle\).
Problem 6
Every complex number \(a+b \mathbf{i}\) may be represented as a point in the complex plane. The unit circle in the complex plane consists of all the complex numbers whose distance from the origin is 1 ; thus, clearly, the unit circle consists of all the complex numbers which can be written in the form $$ \cos x+\mathbf{i} \sin x $$ for some real number \(x\). Prove that \(g(x)=\) cis \(\pi x\) is a homomorphism from \(\mathbb{R}\) onto \(T\), with kernel \(\mathbb{Z}\).
Problem 6
Let \(G\) be a group; let \(H\) and \(K\) be subgroups of \(G\), with \(H\) a normal subgroup of \(G\). Prove the following : By the FHT, \(H /(H \cap K) \cong H K / H\). (This is referred to as the first isomorphism theorem.)
Problem 7
Every complex number \(a+b \mathbf{i}\) may be represented as a point in the complex plane. The unit circle in the complex plane consists of all the complex numbers whose distance from the origin is 1 ; thus, clearly, the unit circle consists of all the complex numbers which can be written in the form $$ \cos x+\mathbf{i} \sin x $$ for some real number \(x\). Conclude that \(T \cong \mathbb{R} / \mathbb{Z}\).
