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Question Number 134523 by mohammad17 last updated on 04/Mar/21

let a,b be any two elements in group G  such that ∣a∣=4 ,∣b∣=2 and a^3 b=ab  find ∣ab∣ ?  help me sir please

$${let}\:{a},{b}\:{be}\:{any}\:{two}\:{elements}\:{in}\:{group}\:{G} \\ $$$${such}\:{that}\:\mid{a}\mid=\mathrm{4}\:,\mid{b}\mid=\mathrm{2}\:{and}\:{a}^{\mathrm{3}} {b}={ab} \\ $$$${find}\:\mid{ab}\mid\:? \\ $$$${help}\:{me}\:{sir}\:{please} \\ $$

Commented by kaivan.ahmadi last updated on 04/Mar/21

a^4 =b^2 =e  a^3 b=ab⇒a^4 b^2 =a^2 b^2 ⇒e=a^2 ⇒∣a∣=2  so the hypothesis is not true.

$${a}^{\mathrm{4}} ={b}^{\mathrm{2}} ={e} \\ $$$${a}^{\mathrm{3}} {b}={ab}\Rightarrow{a}^{\mathrm{4}} {b}^{\mathrm{2}} ={a}^{\mathrm{2}} {b}^{\mathrm{2}} \Rightarrow{e}={a}^{\mathrm{2}} \Rightarrow\mid{a}\mid=\mathrm{2} \\ $$$${so}\:{the}\:{hypothesis}\:{is}\:{not}\:{true}. \\ $$$$ \\ $$

Commented by kaivan.ahmadi last updated on 04/Mar/21

if  ∣a∣=4 and ∣b∣=2 then we have    ∣ab∣=((∣a∣.∣b∣)/((∣a∣,∣b∣)))=((4×2)/2)=4

$${if}\:\:\mid{a}\mid=\mathrm{4}\:{and}\:\mid{b}\mid=\mathrm{2}\:{then}\:{we}\:{have} \\ $$$$ \\ $$$$\mid{ab}\mid=\frac{\mid{a}\mid.\mid{b}\mid}{\left(\mid{a}\mid,\mid{b}\mid\right)}=\frac{\mathrm{4}×\mathrm{2}}{\mathrm{2}}=\mathrm{4} \\ $$

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