Question Number 192746 by Mastermind last updated on 26/May/23
$$\mathrm{Let}\:\mathrm{G}\:\mathrm{be}\:\mathrm{the}\:\mathrm{group}\:\left(\left\{\mathrm{1},\:\imath,\:−\mathrm{1},\:−\imath\right\},\:\centerdot\right) \\ $$$$\mathrm{and}\:\mathrm{let}\:\mathrm{H}\:\leqslant\:\left(\underset{−} {+}\mathrm{1},\:\centerdot\right),\:\mathrm{show}\:\mathrm{that} \\ $$$$\theta:\mathrm{G}\rightarrow\mathrm{H}\:\mathrm{is}\:\mathrm{an}\:\mathrm{Isomorphism}. \\ $$$$ \\ $$$$\mathrm{Hello}! \\ $$
Answered by aleks041103 last updated on 27/May/23
$${If}\:{I}\:{understood}\:{the}\:{question}\:{correctly},\:{we}\: \\ $$$${need}\:{to}\:{show}\:{that}\:{G}\:{is}\:{iomorphic}\:{to}\:{a}\:{subgroup} \\ $$$${of}\:{the}\:{group}\:\left(\left\{+\mathrm{1},−\mathrm{1}\right\},\centerdot\right). \\ $$$${But}\:{G}\cong{C}_{\mathrm{4}} \:{and}\:\left(\left\{+\mathrm{1},−\mathrm{1}\right\},\centerdot\right)\cong{C}_{\mathrm{2}} . \\ $$$$\Rightarrow\forall{H}\leq{C}_{\mathrm{2}} ,\:\mid{H}\mid\in\left\{\mathrm{1},\mathrm{2}\right\} \\ $$$${while}\:\mid{G}\mid=\mathrm{4} \\ $$$${if}\:{H}\cong{G},\:{then}\:\mid{H}\mid=\mid{G}\mid=\mathrm{4}\notin\left\{\mathrm{1},\mathrm{2}\right\} \\ $$$$ \\ $$$${what}\:{you}'{re}\:{asking}\:{is}\:{not}\:{true}\:{and}\:{therefore} \\ $$$${impossible}\:{to}\:{prove}. \\ $$