GATE 2012 MA Group Theory

Question

Comments

@Mk Utkarsh @srestha @AMALJITH

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Ans will be $14$

because here $\left \{ x^{7},y^{2},e \right \}$ all are subgroup of trivial group

Also, $yx=x^{4}y$

$e=x^{-1}.y^{-1}.x^{4}.y$

$e=x^{3}$

$e^{14}=\left ( x^{3} \right )^{14}=e$

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use property order of an element divides the order of group(holds for subgroup also)

since x,y,e are element of subgroup and given that order(x)=7 and order(y)=2 so least possible subgroup size will be LCM(2,7)=14