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I have to find the conjugacy classes of $Q_8$, for which i proceeded as follows:

$Z(Q_8)$={${-1,1}$}

=> $Cl(1)$={${1}$} and $Cl(-1)$={${-1}$}

For all $x,y\in Q_8$ i proved that,

$(-x)y(-x)^{-1}$ = $-1.x.y.(-1.x)^{-1}$ = $-1.x.y.x^{-1}.(-1)^{-1}$ = $-1.x.y.x^{-1}.-1$ = $x.y.x^{-1}$

Using this I found the conjugacy classes of all $x$ such that $x \notin Z(Q_8)$:

$Cl(i)$= {$i.i.i^{-1}=(-i).i.(-i)^{-1}$, $j.i.j^{-1}=(-j).i.(-j)^{-1}$ ,$k.i.k^{-1}=(-k).i.(-k)^{-1}$}

={$i$,$-i$,$-i$} = {$i$,$-i$}

Similarly,

$Cl(j)$ = {$j$,$-j$}

$Cl(k)$ = {$k$,$-k$}

Is this way of doing it correct ?

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Yes. ${}{}{}{}{}{}$ –  Pedro Tamaroff Apr 3 at 3:48

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