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Determine the smallest symmetric group for this condition

I'm having trouble finding a subgroup of $SL_2(F_5)$ isomorphic to $H$, with $H$ generated by $x^4=y^3=1$, $xy=y^2x$.

My first thought went to upper triangular matrices, but that seems like going in the dark... Any hints?

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marked as duplicate by Gerry Myerson, Cameron Buie, martini, TMM, Noah Snyder Nov 14 '12 at 11:09

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Do you mean $H$ is the group $\langle x,y |x^4=y^3=1, xy=y^2x\rangle$? –  Chris Eagle Nov 13 '12 at 18:35
    
@ChrisEagle Yes I do. –  Benjamin Lu Nov 13 '12 at 18:37
    
@ChrisEagle Could you give an example of such an element? –  Benjamin Lu Nov 13 '12 at 19:16
    
Not offhand, no. –  Chris Eagle Nov 13 '12 at 19:21
    
@ChrisEagle Ok, well thanks anyway. –  Benjamin Lu Nov 13 '12 at 19:28
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1 Answer 1

Elements of order 3 have minimal polynomial $x^2+x+1$, so they have trace $-1$. In fact ${\rm SL}_2(5)$ has a unique conjugacy class of such elements, and so you can choose any such element for $y$, say $y=\left(\begin{array}{rr}0&1\\-1&-1\\ \end{array}\right)$.

Elements of order 4 have minimal polynomial $x^2+1$ an hence trace $0$. So you could let $x = \left(\begin{array}{rr}a&b\\c&-a\\ \end{array}\right)$,

where $a^2+bc=-1$, and then plug $x$ and $y$ into the equation $xy=y^2x$ and solve for $a,b,c$.

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