Question Number 134523 by mohammad17 last updated on 04/Mar/21
$${let}\:{a},{b}\:{be}\:{any}\:{two}\:{elements}\:{in}\:{group}\:{G} \\ $$$${such}\:{that}\:\mid{a}\mid=\mathrm{4}\:,\mid{b}\mid=\mathrm{2}\:{and}\:{a}^{\mathrm{3}} {b}={ab} \\ $$$${find}\:\mid{ab}\mid\:? \\ $$$${help}\:{me}\:{sir}\:{please} \\ $$
Commented by kaivan.ahmadi last updated on 04/Mar/21
$${a}^{\mathrm{4}} ={b}^{\mathrm{2}} ={e} \\ $$$${a}^{\mathrm{3}} {b}={ab}\Rightarrow{a}^{\mathrm{4}} {b}^{\mathrm{2}} ={a}^{\mathrm{2}} {b}^{\mathrm{2}} \Rightarrow{e}={a}^{\mathrm{2}} \Rightarrow\mid{a}\mid=\mathrm{2} \\ $$$${so}\:{the}\:{hypothesis}\:{is}\:{not}\:{true}. \\ $$$$ \\ $$
Commented by kaivan.ahmadi last updated on 04/Mar/21
$${if}\:\:\mid{a}\mid=\mathrm{4}\:{and}\:\mid{b}\mid=\mathrm{2}\:{then}\:{we}\:{have} \\ $$$$ \\ $$$$\mid{ab}\mid=\frac{\mid{a}\mid.\mid{b}\mid}{\left(\mid{a}\mid,\mid{b}\mid\right)}=\frac{\mathrm{4}×\mathrm{2}}{\mathrm{2}}=\mathrm{4} \\ $$