Question

# $\left. \begin{array} { l } { \text { Let } H = \{ a + b i | a , b \in \mathbf { R } , a ^ { 2 } + b ^ { 2 } = 1 \} . \text { Prove or disprove that } } \\ { H \text { is a subgroup of C's under multiplication. Describe the elements } } \\ { \text { of } H \text { geometrically. } } \end{array} \right.$

Solution

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Notice that for complex number $a+bi$, expression $a^2 + b^2$ is actually square of it's modulus $|a+bi|$.

We know that modulus is multiplicative, i.e. for two complex numbers $a+bi$ and $c+di$ we have:

$|(a+bi)(c+di)| = |a+bi|\cdot |c+di|$

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